/
multilinear.lean
1629 lines (1375 loc) · 78 KB
/
multilinear.lean
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import analysis.normed_space.operator_norm
import topology.algebra.module.multilinear
/-!
# Operator norm on the space of continuous multilinear maps
When `f` is a continuous multilinear map in finitely many variables, we define its norm `‖f‖` as the
smallest number such that `‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖` for all `m`.
We show that it is indeed a norm, and prove its basic properties.
## Main results
Let `f` be a multilinear map in finitely many variables.
* `exists_bound_of_continuous` asserts that, if `f` is continuous, then there exists `C > 0`
with `‖f m‖ ≤ C * ∏ i, ‖m i‖` for all `m`.
* `continuous_of_bound`, conversely, asserts that this bound implies continuity.
* `mk_continuous` constructs the associated continuous multilinear map.
Let `f` be a continuous multilinear map in finitely many variables.
* `‖f‖` is its norm, i.e., the smallest number such that `‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖` for
all `m`.
* `le_op_norm f m` asserts the fundamental inequality `‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖`.
* `norm_image_sub_le f m₁ m₂` gives a control of the difference `f m₁ - f m₂` in terms of
`‖f‖` and `‖m₁ - m₂‖`.
We also register isomorphisms corresponding to currying or uncurrying variables, transforming a
continuous multilinear function `f` in `n+1` variables into a continuous linear function taking
values in continuous multilinear functions in `n` variables, and also into a continuous multilinear
function in `n` variables taking values in continuous linear functions. These operations are called
`f.curry_left` and `f.curry_right` respectively (with inverses `f.uncurry_left` and
`f.uncurry_right`). They induce continuous linear equivalences between spaces of
continuous multilinear functions in `n+1` variables and spaces of continuous linear functions into
continuous multilinear functions in `n` variables (resp. continuous multilinear functions in `n`
variables taking values in continuous linear functions), called respectively
`continuous_multilinear_curry_left_equiv` and `continuous_multilinear_curry_right_equiv`.
## Implementation notes
We mostly follow the API (and the proofs) of `operator_norm.lean`, with the additional complexity
that we should deal with multilinear maps in several variables. The currying/uncurrying
constructions are based on those in `multilinear.lean`.
From the mathematical point of view, all the results follow from the results on operator norm in
one variable, by applying them to one variable after the other through currying. However, this
is only well defined when there is an order on the variables (for instance on `fin n`) although
the final result is independent of the order. While everything could be done following this
approach, it turns out that direct proofs are easier and more efficient.
-/
noncomputable theory
open_locale big_operators nnreal
open finset metric
local attribute [instance, priority 1001]
add_comm_group.to_add_comm_monoid normed_add_comm_group.to_add_comm_group normed_space.to_module'
/-!
### Type variables
We use the following type variables in this file:
* `𝕜` : a `nontrivially_normed_field`;
* `ι`, `ι'` : finite index types with decidable equality;
* `E`, `E₁` : families of normed vector spaces over `𝕜` indexed by `i : ι`;
* `E'` : a family of normed vector spaces over `𝕜` indexed by `i' : ι'`;
* `Ei` : a family of normed vector spaces over `𝕜` indexed by `i : fin (nat.succ n)`;
* `G`, `G'` : normed vector spaces over `𝕜`.
-/
universes u v v' wE wE₁ wE' wEi wG wG'
variables {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {n : ℕ}
{E : ι → Type wE} {E₁ : ι → Type wE₁} {E' : ι' → Type wE'} {Ei : fin n.succ → Type wEi}
{G : Type wG} {G' : Type wG'}
[fintype ι] [fintype ι'] [nontrivially_normed_field 𝕜]
[Π i, normed_add_comm_group (E i)] [Π i, normed_space 𝕜 (E i)]
[Π i, normed_add_comm_group (E₁ i)] [Π i, normed_space 𝕜 (E₁ i)]
[Π i, normed_add_comm_group (E' i)] [Π i, normed_space 𝕜 (E' i)]
[Π i, normed_add_comm_group (Ei i)] [Π i, normed_space 𝕜 (Ei i)]
[normed_add_comm_group G] [normed_space 𝕜 G] [normed_add_comm_group G'] [normed_space 𝕜 G']
/-!
### Continuity properties of multilinear maps
We relate continuity of multilinear maps to the inequality `‖f m‖ ≤ C * ∏ i, ‖m i‖`, in
both directions. Along the way, we prove useful bounds on the difference `‖f m₁ - f m₂‖`.
-/
namespace multilinear_map
variable (f : multilinear_map 𝕜 E G)
/-- If a multilinear map in finitely many variables on normed spaces satisfies the inequality
`‖f m‖ ≤ C * ∏ i, ‖m i‖` on a shell `ε i / ‖c i‖ < ‖m i‖ < ε i` for some positive numbers `ε i`
and elements `c i : 𝕜`, `1 < ‖c i‖`, then it satisfies this inequality for all `m`. -/
lemma bound_of_shell {ε : ι → ℝ} {C : ℝ} (hε : ∀ i, 0 < ε i) {c : ι → 𝕜} (hc : ∀ i, 1 < ‖c i‖)
(hf : ∀ m : Π i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
(m : Π i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ :=
begin
rcases em (∃ i, m i = 0) with ⟨i, hi⟩|hm; [skip, push_neg at hm],
{ simp [f.map_coord_zero i hi, prod_eq_zero (mem_univ i), hi] },
choose δ hδ0 hδm_lt hle_δm hδinv using λ i, rescale_to_shell (hc i) (hε i) (hm i),
have hδ0 : 0 < ∏ i, ‖δ i‖, from prod_pos (λ i _, norm_pos_iff.2 (hδ0 i)),
simpa [map_smul_univ, norm_smul, prod_mul_distrib, mul_left_comm C, mul_le_mul_left hδ0]
using hf (λ i, δ i • m i) hle_δm hδm_lt,
end
/-- If a multilinear map in finitely many variables on normed spaces is continuous, then it
satisfies the inequality `‖f m‖ ≤ C * ∏ i, ‖m i‖`, for some `C` which can be chosen to be
positive. -/
theorem exists_bound_of_continuous (hf : continuous f) :
∃ (C : ℝ), 0 < C ∧ (∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :=
begin
casesI is_empty_or_nonempty ι,
{ refine ⟨‖f 0‖ + 1, add_pos_of_nonneg_of_pos (norm_nonneg _) zero_lt_one, λ m, _⟩,
obtain rfl : m = 0, from funext (is_empty.elim ‹_›),
simp [univ_eq_empty, zero_le_one] },
obtain ⟨ε : ℝ, ε0 : 0 < ε, hε : ∀ m : Π i, E i, ‖m - 0‖ < ε → ‖f m - f 0‖ < 1⟩ :=
normed_add_comm_group.tendsto_nhds_nhds.1 (hf.tendsto 0) 1 zero_lt_one,
simp only [sub_zero, f.map_zero] at hε,
rcases normed_field.exists_one_lt_norm 𝕜 with ⟨c, hc⟩,
have : 0 < (‖c‖ / ε) ^ fintype.card ι, from pow_pos (div_pos (zero_lt_one.trans hc) ε0) _,
refine ⟨_, this, _⟩,
refine f.bound_of_shell (λ _, ε0) (λ _, hc) (λ m hcm hm, _),
refine (hε m ((pi_norm_lt_iff ε0).2 hm)).le.trans _,
rw [← div_le_iff' this, one_div, ← inv_pow, inv_div, fintype.card, ← prod_const],
exact prod_le_prod (λ _ _, div_nonneg ε0.le (norm_nonneg _)) (λ i _, hcm i)
end
/-- If `f` satisfies a boundedness property around `0`, one can deduce a bound on `f m₁ - f m₂`
using the multilinearity. Here, we give a precise but hard to use version. See
`norm_image_sub_le_of_bound` for a less precise but more usable version. The bound reads
`‖f m - f m'‖ ≤
C * ‖m 1 - m' 1‖ * max ‖m 2‖ ‖m' 2‖ * max ‖m 3‖ ‖m' 3‖ * ... * max ‖m n‖ ‖m' n‖ + ...`,
where the other terms in the sum are the same products where `1` is replaced by any `i`. -/
lemma norm_image_sub_le_of_bound' [decidable_eq ι] {C : ℝ} (hC : 0 ≤ C)
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤
C * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ :=
begin
have A : ∀(s : finset ι), ‖f m₁ - f (s.piecewise m₂ m₁)‖
≤ C * ∑ i in s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖,
{ refine finset.induction (by simp) _,
assume i s his Hrec,
have I : ‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖
≤ C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖,
{ have A : ((insert i s).piecewise m₂ m₁)
= function.update (s.piecewise m₂ m₁) i (m₂ i) := s.piecewise_insert _ _ _,
have B : s.piecewise m₂ m₁ = function.update (s.piecewise m₂ m₁) i (m₁ i),
{ ext j,
by_cases h : j = i,
{ rw h, simp [his] },
{ simp [h] } },
rw [B, A, ← f.map_sub],
apply le_trans (H _) (mul_le_mul_of_nonneg_left _ hC),
refine prod_le_prod (λj hj, norm_nonneg _) (λj hj, _),
by_cases h : j = i,
{ rw h, simp },
{ by_cases h' : j ∈ s;
simp [h', h, le_refl] } },
calc ‖f m₁ - f ((insert i s).piecewise m₂ m₁)‖ ≤
‖f m₁ - f (s.piecewise m₂ m₁)‖ + ‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖ :
by { rw [← dist_eq_norm, ← dist_eq_norm, ← dist_eq_norm], exact dist_triangle _ _ _ }
... ≤ (C * ∑ i in s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖)
+ C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ :
add_le_add Hrec I
... = C * ∑ i in insert i s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ :
by simp [his, add_comm, left_distrib] },
convert A univ,
simp
end
/-- If `f` satisfies a boundedness property around `0`, one can deduce a bound on `f m₁ - f m₂`
using the multilinearity. Here, we give a usable but not very precise version. See
`norm_image_sub_le_of_bound'` for a more precise but less usable version. The bound is
`‖f m - f m'‖ ≤ C * card ι * ‖m - m'‖ * (max ‖m‖ ‖m'‖) ^ (card ι - 1)`. -/
lemma norm_image_sub_le_of_bound {C : ℝ} (hC : 0 ≤ C)
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤ C * (fintype.card ι) * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1) * ‖m₁ - m₂‖ :=
begin
letI := classical.dec_eq ι,
have A : ∀ (i : ι), ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖)
≤ ‖m₁ - m₂‖ * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1),
{ assume i,
calc ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖)
≤ ∏ j : ι, function.update (λ j, max ‖m₁‖ ‖m₂‖) i (‖m₁ - m₂‖) j :
begin
apply prod_le_prod,
{ assume j hj, by_cases h : j = i; simp [h, norm_nonneg] },
{ assume j hj,
by_cases h : j = i,
{ rw h, simp, exact norm_le_pi_norm (m₁ - m₂) i },
{ simp [h, max_le_max, norm_le_pi_norm (_ : Π i, E i)] } }
end
... = ‖m₁ - m₂‖ * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1) :
by { rw prod_update_of_mem (finset.mem_univ _), simp [card_univ_diff] } },
calc
‖f m₁ - f m₂‖
≤ C * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ :
f.norm_image_sub_le_of_bound' hC H m₁ m₂
... ≤ C * ∑ i, ‖m₁ - m₂‖ * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1) :
mul_le_mul_of_nonneg_left (sum_le_sum (λi hi, A i)) hC
... = C * (fintype.card ι) * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1) * ‖m₁ - m₂‖ :
by { rw [sum_const, card_univ, nsmul_eq_mul], ring }
end
/-- If a multilinear map satisfies an inequality `‖f m‖ ≤ C * ∏ i, ‖m i‖`, then it is
continuous. -/
theorem continuous_of_bound (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :
continuous f :=
begin
let D := max C 1,
have D_pos : 0 ≤ D := le_trans zero_le_one (le_max_right _ _),
replace H : ∀ m, ‖f m‖ ≤ D * ∏ i, ‖m i‖,
{ assume m,
apply le_trans (H m) (mul_le_mul_of_nonneg_right (le_max_left _ _) _),
exact prod_nonneg (λ(i : ι) hi, norm_nonneg (m i)) },
refine continuous_iff_continuous_at.2 (λm, _),
refine continuous_at_of_locally_lipschitz zero_lt_one
(D * (fintype.card ι) * (‖m‖ + 1) ^ (fintype.card ι - 1)) (λm' h', _),
rw [dist_eq_norm, dist_eq_norm],
have : 0 ≤ (max ‖m'‖ ‖m‖), by simp,
have : (max ‖m'‖ ‖m‖) ≤ ‖m‖ + 1,
by simp [zero_le_one, norm_le_of_mem_closed_ball (le_of_lt h'), -add_comm],
calc
‖f m' - f m‖
≤ D * (fintype.card ι) * (max ‖m'‖ ‖m‖) ^ (fintype.card ι - 1) * ‖m' - m‖ :
f.norm_image_sub_le_of_bound D_pos H m' m
... ≤ D * (fintype.card ι) * (‖m‖ + 1) ^ (fintype.card ι - 1) * ‖m' - m‖ :
by apply_rules [mul_le_mul_of_nonneg_right, mul_le_mul_of_nonneg_left, mul_nonneg,
norm_nonneg, nat.cast_nonneg, pow_le_pow_of_le_left]
end
/-- Constructing a continuous multilinear map from a multilinear map satisfying a boundedness
condition. -/
def mk_continuous (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :
continuous_multilinear_map 𝕜 E G :=
{ cont := f.continuous_of_bound C H, ..f }
@[simp] lemma coe_mk_continuous (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :
⇑(f.mk_continuous C H) = f :=
rfl
/-- Given a multilinear map in `n` variables, if one restricts it to `k` variables putting `z` on
the other coordinates, then the resulting restricted function satisfies an inequality
`‖f.restr v‖ ≤ C * ‖z‖^(n-k) * Π ‖v i‖` if the original function satisfies `‖f v‖ ≤ C * Π ‖v i‖`. -/
lemma restr_norm_le {k n : ℕ} (f : (multilinear_map 𝕜 (λ i : fin n, G) G' : _))
(s : finset (fin n)) (hk : s.card = k) (z : G) {C : ℝ}
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (v : fin k → G) :
‖f.restr s hk z v‖ ≤ C * ‖z‖ ^ (n - k) * ∏ i, ‖v i‖ :=
begin
rw [mul_right_comm, mul_assoc],
convert H _ using 2,
simp only [apply_dite norm, fintype.prod_dite, prod_const (‖z‖), finset.card_univ,
fintype.card_of_subtype sᶜ (λ x, mem_compl), card_compl, fintype.card_fin, hk, mk_coe,
← (s.order_iso_of_fin hk).symm.bijective.prod_comp (λ x, ‖v x‖)],
refl
end
end multilinear_map
/-!
### Continuous multilinear maps
We define the norm `‖f‖` of a continuous multilinear map `f` in finitely many variables as the
smallest number such that `‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖` for all `m`. We show that this
defines a normed space structure on `continuous_multilinear_map 𝕜 E G`.
-/
namespace continuous_multilinear_map
variables (c : 𝕜) (f g : continuous_multilinear_map 𝕜 E G) (m : Πi, E i)
theorem bound : ∃ (C : ℝ), 0 < C ∧ (∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :=
f.to_multilinear_map.exists_bound_of_continuous f.2
open real
/-- The operator norm of a continuous multilinear map is the inf of all its bounds. -/
def op_norm := Inf {c | 0 ≤ (c : ℝ) ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖}
instance has_op_norm : has_norm (continuous_multilinear_map 𝕜 E G) := ⟨op_norm⟩
/-- An alias of `continuous_multilinear_map.has_op_norm` with non-dependent types to help typeclass
search. -/
instance has_op_norm' : has_norm (continuous_multilinear_map 𝕜 (λ (i : ι), G) G') :=
continuous_multilinear_map.has_op_norm
lemma norm_def : ‖f‖ = Inf {c | 0 ≤ (c : ℝ) ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} := rfl
-- So that invocations of `le_cInf` make sense: we show that the set of
-- bounds is nonempty and bounded below.
lemma bounds_nonempty {f : continuous_multilinear_map 𝕜 E G} :
∃ c, c ∈ {c | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} :=
let ⟨M, hMp, hMb⟩ := f.bound in ⟨M, le_of_lt hMp, hMb⟩
lemma bounds_bdd_below {f : continuous_multilinear_map 𝕜 E G} :
bdd_below {c | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} :=
⟨0, λ _ ⟨hn, _⟩, hn⟩
lemma op_norm_nonneg : 0 ≤ ‖f‖ :=
le_cInf bounds_nonempty (λ _ ⟨hx, _⟩, hx)
/-- The fundamental property of the operator norm of a continuous multilinear map:
`‖f m‖` is bounded by `‖f‖` times the product of the `‖m i‖`. -/
theorem le_op_norm : ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ :=
begin
have A : 0 ≤ ∏ i, ‖m i‖ := prod_nonneg (λj hj, norm_nonneg _),
cases A.eq_or_lt with h hlt,
{ rcases prod_eq_zero_iff.1 h.symm with ⟨i, _, hi⟩,
rw norm_eq_zero at hi,
have : f m = 0 := f.map_coord_zero i hi,
rw [this, norm_zero],
exact mul_nonneg (op_norm_nonneg f) A },
{ rw [← div_le_iff hlt],
apply le_cInf bounds_nonempty,
rintro c ⟨_, hc⟩, rw [div_le_iff hlt], apply hc }
end
theorem le_of_op_norm_le {C : ℝ} (h : ‖f‖ ≤ C) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ :=
(f.le_op_norm m).trans $ mul_le_mul_of_nonneg_right h (prod_nonneg $ λ i _, norm_nonneg (m i))
lemma ratio_le_op_norm : ‖f m‖ / ∏ i, ‖m i‖ ≤ ‖f‖ :=
div_le_of_nonneg_of_le_mul (prod_nonneg $ λ i _, norm_nonneg _) (op_norm_nonneg _) (f.le_op_norm m)
/-- The image of the unit ball under a continuous multilinear map is bounded. -/
lemma unit_le_op_norm (h : ‖m‖ ≤ 1) : ‖f m‖ ≤ ‖f‖ :=
calc
‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ : f.le_op_norm m
... ≤ ‖f‖ * ∏ i : ι, 1 :
mul_le_mul_of_nonneg_left (prod_le_prod (λi hi, norm_nonneg _)
(λi hi, le_trans (norm_le_pi_norm (_ : Π i, E i) _) h)) (op_norm_nonneg f)
... = ‖f‖ : by simp
/-- If one controls the norm of every `f x`, then one controls the norm of `f`. -/
lemma op_norm_le_bound {M : ℝ} (hMp: 0 ≤ M) (hM : ∀ m, ‖f m‖ ≤ M * ∏ i, ‖m i‖) :
‖f‖ ≤ M :=
cInf_le bounds_bdd_below ⟨hMp, hM⟩
/-- The operator norm satisfies the triangle inequality. -/
theorem op_norm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖ :=
cInf_le bounds_bdd_below
⟨add_nonneg (op_norm_nonneg _) (op_norm_nonneg _), λ x, by { rw add_mul,
exact norm_add_le_of_le (le_op_norm _ _) (le_op_norm _ _) }⟩
lemma op_norm_zero : ‖(0 : continuous_multilinear_map 𝕜 E G)‖ = 0 :=
(op_norm_nonneg _).antisymm' $ op_norm_le_bound 0 le_rfl $ λ m, by simp
/-- A continuous linear map is zero iff its norm vanishes. -/
theorem op_norm_zero_iff : ‖f‖ = 0 ↔ f = 0 :=
⟨λ h, by { ext m, simpa [h] using f.le_op_norm m }, by { rintro rfl, exact op_norm_zero }⟩
section
variables {𝕜' : Type*} [normed_field 𝕜'] [normed_space 𝕜' G] [smul_comm_class 𝕜 𝕜' G]
lemma op_norm_smul_le (c : 𝕜') : ‖c • f‖ ≤ ‖c‖ * ‖f‖ :=
(c • f).op_norm_le_bound
(mul_nonneg (norm_nonneg _) (op_norm_nonneg _))
begin
intro m,
erw [norm_smul, mul_assoc],
exact mul_le_mul_of_nonneg_left (le_op_norm _ _) (norm_nonneg _)
end
lemma op_norm_neg : ‖-f‖ = ‖f‖ := by { rw norm_def, apply congr_arg, ext, simp }
/-- Continuous multilinear maps themselves form a normed space with respect to
the operator norm. -/
instance normed_add_comm_group : normed_add_comm_group (continuous_multilinear_map 𝕜 E G) :=
add_group_norm.to_normed_add_comm_group
{ to_fun := norm,
map_zero' := op_norm_zero,
neg' := op_norm_neg,
add_le' := op_norm_add_le,
eq_zero_of_map_eq_zero' := λ f, f.op_norm_zero_iff.1 }
/-- An alias of `continuous_multilinear_map.normed_add_comm_group` with non-dependent types to help
typeclass search. -/
instance normed_add_comm_group' :
normed_add_comm_group (continuous_multilinear_map 𝕜 (λ i : ι, G) G') :=
continuous_multilinear_map.normed_add_comm_group
instance normed_space : normed_space 𝕜' (continuous_multilinear_map 𝕜 E G) :=
⟨λ c f, f.op_norm_smul_le c⟩
/-- An alias of `continuous_multilinear_map.normed_space` with non-dependent types to help typeclass
search. -/
instance normed_space' : normed_space 𝕜' (continuous_multilinear_map 𝕜 (λ i : ι, G') G) :=
continuous_multilinear_map.normed_space
theorem le_op_norm_mul_prod_of_le {b : ι → ℝ} (hm : ∀ i, ‖m i‖ ≤ b i) : ‖f m‖ ≤ ‖f‖ * ∏ i, b i :=
(f.le_op_norm m).trans $ mul_le_mul_of_nonneg_left
(prod_le_prod (λ _ _, norm_nonneg _) (λ i _, hm i)) (norm_nonneg f)
theorem le_op_norm_mul_pow_card_of_le {b : ℝ} (hm : ∀ i, ‖m i‖ ≤ b) :
‖f m‖ ≤ ‖f‖ * b ^ fintype.card ι :=
by simpa only [prod_const] using f.le_op_norm_mul_prod_of_le m hm
theorem le_op_norm_mul_pow_of_le {Ei : fin n → Type*} [Π i, normed_add_comm_group (Ei i)]
[Π i, normed_space 𝕜 (Ei i)] (f : continuous_multilinear_map 𝕜 Ei G) (m : Π i, Ei i)
{b : ℝ} (hm : ‖m‖ ≤ b) :
‖f m‖ ≤ ‖f‖ * b ^ n :=
by simpa only [fintype.card_fin]
using f.le_op_norm_mul_pow_card_of_le m (λ i, (norm_le_pi_norm m i).trans hm)
/-- The fundamental property of the operator norm of a continuous multilinear map:
`‖f m‖` is bounded by `‖f‖` times the product of the `‖m i‖`, `nnnorm` version. -/
theorem le_op_nnnorm : ‖f m‖₊ ≤ ‖f‖₊ * ∏ i, ‖m i‖₊ :=
nnreal.coe_le_coe.1 $ by { push_cast, exact f.le_op_norm m }
theorem le_of_op_nnnorm_le {C : ℝ≥0} (h : ‖f‖₊ ≤ C) : ‖f m‖₊ ≤ C * ∏ i, ‖m i‖₊ :=
(f.le_op_nnnorm m).trans $ mul_le_mul' h le_rfl
lemma op_norm_prod (f : continuous_multilinear_map 𝕜 E G) (g : continuous_multilinear_map 𝕜 E G') :
‖f.prod g‖ = max (‖f‖) (‖g‖) :=
le_antisymm
(op_norm_le_bound _ (norm_nonneg (f, g)) (λ m,
have H : 0 ≤ ∏ i, ‖m i‖, from prod_nonneg $ λ _ _, norm_nonneg _,
by simpa only [prod_apply, prod.norm_def, max_mul_of_nonneg, H]
using max_le_max (f.le_op_norm m) (g.le_op_norm m))) $
max_le
(f.op_norm_le_bound (norm_nonneg _) $ λ m, (le_max_left _ _).trans ((f.prod g).le_op_norm _))
(g.op_norm_le_bound (norm_nonneg _) $ λ m, (le_max_right _ _).trans ((f.prod g).le_op_norm _))
lemma norm_pi {ι' : Type v'} [fintype ι'] {E' : ι' → Type wE'} [Π i', normed_add_comm_group (E' i')]
[Π i', normed_space 𝕜 (E' i')] (f : Π i', continuous_multilinear_map 𝕜 E (E' i')) :
‖pi f‖ = ‖f‖ :=
begin
apply le_antisymm,
{ refine (op_norm_le_bound _ (norm_nonneg f) (λ m, _)),
dsimp,
rw pi_norm_le_iff_of_nonneg,
exacts [λ i, (f i).le_of_op_norm_le m (norm_le_pi_norm f i),
mul_nonneg (norm_nonneg f) (prod_nonneg $ λ _ _, norm_nonneg _)] },
{ refine (pi_norm_le_iff_of_nonneg (norm_nonneg _)).2 (λ i, _),
refine (op_norm_le_bound _ (norm_nonneg _) (λ m, _)),
refine le_trans _ ((pi f).le_op_norm m),
convert norm_le_pi_norm (λ j, f j m) i }
end
section
variables (𝕜 G)
@[simp] lemma norm_of_subsingleton [subsingleton ι] [nontrivial G] (i' : ι) :
‖of_subsingleton 𝕜 G i'‖ = 1 :=
begin
apply le_antisymm,
{ refine op_norm_le_bound _ zero_le_one (λ m, _),
rw [fintype.prod_subsingleton _ i', one_mul, of_subsingleton_apply] },
{ obtain ⟨g, hg⟩ := exists_ne (0 : G),
rw ←norm_ne_zero_iff at hg,
have := (of_subsingleton 𝕜 G i').ratio_le_op_norm (λ _, g),
rwa [fintype.prod_subsingleton _ i', of_subsingleton_apply, div_self hg] at this },
end
@[simp] lemma nnnorm_of_subsingleton [subsingleton ι] [nontrivial G] (i' : ι) :
‖of_subsingleton 𝕜 G i'‖₊ = 1 :=
nnreal.eq $ norm_of_subsingleton _ _ _
variables {G} (E)
@[simp] lemma norm_const_of_is_empty [is_empty ι] (x : G) : ‖const_of_is_empty 𝕜 E x‖ = ‖x‖ :=
begin
apply le_antisymm,
{ refine op_norm_le_bound _ (norm_nonneg _) (λ x, _),
rw [fintype.prod_empty, mul_one, const_of_is_empty_apply], },
{ simpa using (const_of_is_empty 𝕜 E x).le_op_norm 0 }
end
@[simp] lemma nnnorm_const_of_is_empty [is_empty ι] (x : G) : ‖const_of_is_empty 𝕜 E x‖₊ = ‖x‖₊ :=
nnreal.eq $ norm_const_of_is_empty _ _ _
end
section
variables (𝕜 E E' G G')
/-- `continuous_multilinear_map.prod` as a `linear_isometry_equiv`. -/
def prodL :
(continuous_multilinear_map 𝕜 E G) × (continuous_multilinear_map 𝕜 E G') ≃ₗᵢ[𝕜]
continuous_multilinear_map 𝕜 E (G × G') :=
{ to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ((continuous_linear_map.fst 𝕜 G G').comp_continuous_multilinear_map f,
(continuous_linear_map.snd 𝕜 G G').comp_continuous_multilinear_map f),
map_add' := λ f g, rfl,
map_smul' := λ c f, rfl,
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl,
norm_map' := λ f, op_norm_prod f.1 f.2 }
/-- `continuous_multilinear_map.pi` as a `linear_isometry_equiv`. -/
def piₗᵢ {ι' : Type v'} [fintype ι'] {E' : ι' → Type wE'} [Π i', normed_add_comm_group (E' i')]
[Π i', normed_space 𝕜 (E' i')] :
@linear_isometry_equiv 𝕜 𝕜 _ _ (ring_hom.id 𝕜) _ _ _
(Π i', continuous_multilinear_map 𝕜 E (E' i')) (continuous_multilinear_map 𝕜 E (Π i, E' i)) _ _
(@pi.module ι' _ 𝕜 _ _ (λ i', infer_instance)) _ :=
{ to_linear_equiv :=
-- note: `pi_linear_equiv` does not unify correctly here, presumably due to issues with dependent
-- typeclass arguments.
{ map_add' := λ f g, rfl,
map_smul' := λ c f, rfl,
.. pi_equiv, },
norm_map' := norm_pi }
end
end
section restrict_scalars
variables {𝕜' : Type*} [nontrivially_normed_field 𝕜'] [normed_algebra 𝕜' 𝕜]
variables [normed_space 𝕜' G] [is_scalar_tower 𝕜' 𝕜 G]
variables [Π i, normed_space 𝕜' (E i)] [∀ i, is_scalar_tower 𝕜' 𝕜 (E i)]
@[simp] lemma norm_restrict_scalars : ‖f.restrict_scalars 𝕜'‖ = ‖f‖ :=
by simp only [norm_def, coe_restrict_scalars]
variable (𝕜')
/-- `continuous_multilinear_map.restrict_scalars` as a `continuous_multilinear_map`. -/
def restrict_scalars_linear :
continuous_multilinear_map 𝕜 E G →L[𝕜'] continuous_multilinear_map 𝕜' E G :=
linear_map.mk_continuous
{ to_fun := restrict_scalars 𝕜',
map_add' := λ m₁ m₂, rfl,
map_smul' := λ c m, rfl } 1 $ λ f, by simp
variable {𝕜'}
lemma continuous_restrict_scalars :
continuous (restrict_scalars 𝕜' : continuous_multilinear_map 𝕜 E G →
continuous_multilinear_map 𝕜' E G) :=
(restrict_scalars_linear 𝕜').continuous
end restrict_scalars
/-- The difference `f m₁ - f m₂` is controlled in terms of `‖f‖` and `‖m₁ - m₂‖`, precise version.
For a less precise but more usable version, see `norm_image_sub_le`. The bound reads
`‖f m - f m'‖ ≤
‖f‖ * ‖m 1 - m' 1‖ * max ‖m 2‖ ‖m' 2‖ * max ‖m 3‖ ‖m' 3‖ * ... * max ‖m n‖ ‖m' n‖ + ...`,
where the other terms in the sum are the same products where `1` is replaced by any `i`.-/
lemma norm_image_sub_le' [decidable_eq ι] (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤
‖f‖ * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ :=
f.to_multilinear_map.norm_image_sub_le_of_bound' (norm_nonneg _) f.le_op_norm _ _
/-- The difference `f m₁ - f m₂` is controlled in terms of `‖f‖` and `‖m₁ - m₂‖`, less precise
version. For a more precise but less usable version, see `norm_image_sub_le'`.
The bound is `‖f m - f m'‖ ≤ ‖f‖ * card ι * ‖m - m'‖ * (max ‖m‖ ‖m'‖) ^ (card ι - 1)`.-/
lemma norm_image_sub_le (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤ ‖f‖ * (fintype.card ι) * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1) * ‖m₁ - m₂‖ :=
f.to_multilinear_map.norm_image_sub_le_of_bound (norm_nonneg _) f.le_op_norm _ _
/-- Applying a multilinear map to a vector is continuous in both coordinates. -/
lemma continuous_eval :
continuous (λ p : continuous_multilinear_map 𝕜 E G × Π i, E i, p.1 p.2) :=
begin
apply continuous_iff_continuous_at.2 (λp, _),
apply continuous_at_of_locally_lipschitz zero_lt_one
((‖p‖ + 1) * (fintype.card ι) * (‖p‖ + 1) ^ (fintype.card ι - 1) + ∏ i, ‖p.2 i‖)
(λq hq, _),
have : 0 ≤ (max ‖q.2‖ ‖p.2‖), by simp,
have : 0 ≤ ‖p‖ + 1 := zero_le_one.trans ((le_add_iff_nonneg_left 1).2 $ norm_nonneg p),
have A : ‖q‖ ≤ ‖p‖ + 1 := norm_le_of_mem_closed_ball hq.le,
have : (max ‖q.2‖ ‖p.2‖) ≤ ‖p‖ + 1 :=
(max_le_max (norm_snd_le q) (norm_snd_le p)).trans (by simp [A, -add_comm, zero_le_one]),
have : ∀ (i : ι), i ∈ univ → 0 ≤ ‖p.2 i‖ := λ i hi, norm_nonneg _,
calc dist (q.1 q.2) (p.1 p.2)
≤ dist (q.1 q.2) (q.1 p.2) + dist (q.1 p.2) (p.1 p.2) : dist_triangle _ _ _
... = ‖q.1 q.2 - q.1 p.2‖ + ‖q.1 p.2 - p.1 p.2‖ : by rw [dist_eq_norm, dist_eq_norm]
... ≤ ‖q.1‖ * (fintype.card ι) * (max ‖q.2‖ ‖p.2‖) ^ (fintype.card ι - 1) * ‖q.2 - p.2‖
+ ‖q.1 - p.1‖ * ∏ i, ‖p.2 i‖ :
add_le_add (norm_image_sub_le _ _ _) ((q.1 - p.1).le_op_norm p.2)
... ≤ (‖p‖ + 1) * (fintype.card ι) * (‖p‖ + 1) ^ (fintype.card ι - 1) * ‖q - p‖
+ ‖q - p‖ * ∏ i, ‖p.2 i‖ :
by apply_rules [add_le_add, mul_le_mul, le_refl, le_trans (norm_fst_le q) A, nat.cast_nonneg,
mul_nonneg, pow_le_pow_of_le_left, pow_nonneg, norm_snd_le (q - p), norm_nonneg,
norm_fst_le (q - p), prod_nonneg]
... = ((‖p‖ + 1) * (fintype.card ι) * (‖p‖ + 1) ^ (fintype.card ι - 1)
+ (∏ i, ‖p.2 i‖)) * dist q p : by { rw dist_eq_norm, ring }
end
lemma continuous_eval_left (m : Π i, E i) :
continuous (λ p : continuous_multilinear_map 𝕜 E G, p m) :=
continuous_eval.comp (continuous_id.prod_mk continuous_const)
lemma has_sum_eval
{α : Type*} {p : α → continuous_multilinear_map 𝕜 E G} {q : continuous_multilinear_map 𝕜 E G}
(h : has_sum p q) (m : Π i, E i) : has_sum (λ a, p a m) (q m) :=
begin
dsimp [has_sum] at h ⊢,
convert ((continuous_eval_left m).tendsto _).comp h,
ext s,
simp
end
lemma tsum_eval {α : Type*} {p : α → continuous_multilinear_map 𝕜 E G} (hp : summable p)
(m : Π i, E i) : (∑' a, p a) m = ∑' a, p a m :=
(has_sum_eval hp.has_sum m).tsum_eq.symm
open_locale topology
open filter
/-- If the target space is complete, the space of continuous multilinear maps with its norm is also
complete. The proof is essentially the same as for the space of continuous linear maps (modulo the
addition of `finset.prod` where needed. The duplication could be avoided by deducing the linear
case from the multilinear case via a currying isomorphism. However, this would mess up imports,
and it is more satisfactory to have the simplest case as a standalone proof. -/
instance [complete_space G] : complete_space (continuous_multilinear_map 𝕜 E G) :=
begin
have nonneg : ∀ (v : Π i, E i), 0 ≤ ∏ i, ‖v i‖ :=
λ v, finset.prod_nonneg (λ i hi, norm_nonneg _),
-- We show that every Cauchy sequence converges.
refine metric.complete_of_cauchy_seq_tendsto (λ f hf, _),
-- We now expand out the definition of a Cauchy sequence,
rcases cauchy_seq_iff_le_tendsto_0.1 hf with ⟨b, b0, b_bound, b_lim⟩,
-- and establish that the evaluation at any point `v : Π i, E i` is Cauchy.
have cau : ∀ v, cauchy_seq (λ n, f n v),
{ assume v,
apply cauchy_seq_iff_le_tendsto_0.2 ⟨λ n, b n * ∏ i, ‖v i‖, λ n, _, _, _⟩,
{ exact mul_nonneg (b0 n) (nonneg v) },
{ assume n m N hn hm,
rw dist_eq_norm,
apply le_trans ((f n - f m).le_op_norm v) _,
exact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v) },
{ simpa using b_lim.mul tendsto_const_nhds } },
-- We assemble the limits points of those Cauchy sequences
-- (which exist as `G` is complete)
-- into a function which we call `F`.
choose F hF using λv, cauchy_seq_tendsto_of_complete (cau v),
-- Next, we show that this `F` is multilinear,
let Fmult : multilinear_map 𝕜 E G :=
{ to_fun := F,
map_add' := λ _ v i x y, begin
resetI,
have A := hF (function.update v i (x + y)),
have B := (hF (function.update v i x)).add (hF (function.update v i y)),
simp at A B,
exact tendsto_nhds_unique A B
end,
map_smul' := λ _ v i c x, begin
resetI,
have A := hF (function.update v i (c • x)),
have B := filter.tendsto.smul (@tendsto_const_nhds _ ℕ _ c _) (hF (function.update v i x)),
simp at A B,
exact tendsto_nhds_unique A B
end },
-- and that `F` has norm at most `(b 0 + ‖f 0‖)`.
have Fnorm : ∀ v, ‖F v‖ ≤ (b 0 + ‖f 0‖) * ∏ i, ‖v i‖,
{ assume v,
have A : ∀ n, ‖f n v‖ ≤ (b 0 + ‖f 0‖) * ∏ i, ‖v i‖,
{ assume n,
apply le_trans ((f n).le_op_norm _) _,
apply mul_le_mul_of_nonneg_right _ (nonneg v),
calc ‖f n‖ = ‖(f n - f 0) + f 0‖ : by { congr' 1, abel }
... ≤ ‖f n - f 0‖ + ‖f 0‖ : norm_add_le _ _
... ≤ b 0 + ‖f 0‖ : begin
apply add_le_add_right,
simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _)
end },
exact le_of_tendsto (hF v).norm (eventually_of_forall A) },
-- Thus `F` is continuous, and we propose that as the limit point of our original Cauchy sequence.
let Fcont := Fmult.mk_continuous _ Fnorm,
use Fcont,
-- Our last task is to establish convergence to `F` in norm.
have : ∀ n, ‖f n - Fcont‖ ≤ b n,
{ assume n,
apply op_norm_le_bound _ (b0 n) (λ v, _),
have A : ∀ᶠ m in at_top, ‖(f n - f m) v‖ ≤ b n * ∏ i, ‖v i‖,
{ refine eventually_at_top.2 ⟨n, λ m hm, _⟩,
apply le_trans ((f n - f m).le_op_norm _) _,
exact mul_le_mul_of_nonneg_right (b_bound n m n le_rfl hm) (nonneg v) },
have B : tendsto (λ m, ‖(f n - f m) v‖) at_top (𝓝 (‖(f n - Fcont) v‖)) :=
tendsto.norm (tendsto_const_nhds.sub (hF v)),
exact le_of_tendsto B A },
erw tendsto_iff_norm_tendsto_zero,
exact squeeze_zero (λ n, norm_nonneg _) this b_lim,
end
end continuous_multilinear_map
/-- If a continuous multilinear map is constructed from a multilinear map via the constructor
`mk_continuous`, then its norm is bounded by the bound given to the constructor if it is
nonnegative. -/
lemma multilinear_map.mk_continuous_norm_le (f : multilinear_map 𝕜 E G) {C : ℝ} (hC : 0 ≤ C)
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : ‖f.mk_continuous C H‖ ≤ C :=
continuous_multilinear_map.op_norm_le_bound _ hC (λm, H m)
/-- If a continuous multilinear map is constructed from a multilinear map via the constructor
`mk_continuous`, then its norm is bounded by the bound given to the constructor if it is
nonnegative. -/
lemma multilinear_map.mk_continuous_norm_le' (f : multilinear_map 𝕜 E G) {C : ℝ}
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : ‖f.mk_continuous C H‖ ≤ max C 0 :=
continuous_multilinear_map.op_norm_le_bound _ (le_max_right _ _) $
λ m, (H m).trans $ mul_le_mul_of_nonneg_right (le_max_left _ _)
(prod_nonneg $ λ _ _, norm_nonneg _)
namespace continuous_multilinear_map
/-- Given a continuous multilinear map `f` on `n` variables (parameterized by `fin n`) and a subset
`s` of `k` of these variables, one gets a new continuous multilinear map on `fin k` by varying
these variables, and fixing the other ones equal to a given value `z`. It is denoted by
`f.restr s hk z`, where `hk` is a proof that the cardinality of `s` is `k`. The implicit
identification between `fin k` and `s` that we use is the canonical (increasing) bijection. -/
def restr {k n : ℕ} (f : (G [×n]→L[𝕜] G' : _)) (s : finset (fin n)) (hk : s.card = k) (z : G) :
G [×k]→L[𝕜] G' :=
(f.to_multilinear_map.restr s hk z).mk_continuous
(‖f‖ * ‖z‖^(n-k)) $ λ v, multilinear_map.restr_norm_le _ _ _ _ f.le_op_norm _
lemma norm_restr {k n : ℕ} (f : G [×n]→L[𝕜] G') (s : finset (fin n)) (hk : s.card = k) (z : G) :
‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k) :=
begin
apply multilinear_map.mk_continuous_norm_le,
exact mul_nonneg (norm_nonneg _) (pow_nonneg (norm_nonneg _) _)
end
section
variables {𝕜 ι} {A : Type*} [normed_comm_ring A] [normed_algebra 𝕜 A]
@[simp]
lemma norm_mk_pi_algebra_le [nonempty ι] :
‖continuous_multilinear_map.mk_pi_algebra 𝕜 ι A‖ ≤ 1 :=
begin
have := λ f, @op_norm_le_bound 𝕜 ι (λ i, A) A _ _ _ _ _ _ f _ zero_le_one,
refine this _ _,
intros m,
simp only [continuous_multilinear_map.mk_pi_algebra_apply, one_mul],
exact norm_prod_le' _ univ_nonempty _,
end
lemma norm_mk_pi_algebra_of_empty [is_empty ι] :
‖continuous_multilinear_map.mk_pi_algebra 𝕜 ι A‖ = ‖(1 : A)‖ :=
begin
apply le_antisymm,
{ have := λ f, @op_norm_le_bound 𝕜 ι (λ i, A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A)),
refine this _ _,
simp, },
{ convert ratio_le_op_norm _ (λ _, (1 : A)),
simp [eq_empty_of_is_empty (univ : finset ι)], },
end
@[simp] lemma norm_mk_pi_algebra [norm_one_class A] :
‖continuous_multilinear_map.mk_pi_algebra 𝕜 ι A‖ = 1 :=
begin
casesI is_empty_or_nonempty ι,
{ simp [norm_mk_pi_algebra_of_empty] },
{ refine le_antisymm norm_mk_pi_algebra_le _,
convert ratio_le_op_norm _ (λ _, 1); [skip, apply_instance],
simp },
end
end
section
variables {𝕜 n} {A : Type*} [normed_ring A] [normed_algebra 𝕜 A]
lemma norm_mk_pi_algebra_fin_succ_le :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 n.succ A‖ ≤ 1 :=
begin
have := λ f, @op_norm_le_bound 𝕜 (fin n.succ) (λ i, A) A _ _ _ _ _ _ f _ zero_le_one,
refine this _ _,
intros m,
simp only [continuous_multilinear_map.mk_pi_algebra_fin_apply, one_mul, list.of_fn_eq_map,
fin.prod_univ_def, multiset.coe_map, multiset.coe_prod],
refine (list.norm_prod_le' _).trans_eq _,
{ rw [ne.def, list.map_eq_nil, list.fin_range_eq_nil],
exact nat.succ_ne_zero _, },
rw list.map_map,
end
lemma norm_mk_pi_algebra_fin_le_of_pos (hn : 0 < n) :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 n A‖ ≤ 1 :=
begin
obtain ⟨n, rfl⟩ := nat.exists_eq_succ_of_ne_zero hn.ne',
exact norm_mk_pi_algebra_fin_succ_le
end
lemma norm_mk_pi_algebra_fin_zero :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 0 A‖ = ‖(1 : A)‖ :=
begin
refine le_antisymm _ _,
{ have := λ f, @op_norm_le_bound 𝕜 (fin 0) (λ i, A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A)),
refine this _ _,
simp, },
{ convert ratio_le_op_norm _ (λ _, (1 : A)),
simp }
end
@[simp] lemma norm_mk_pi_algebra_fin [norm_one_class A] :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 n A‖ = 1 :=
begin
cases n,
{ simp [norm_mk_pi_algebra_fin_zero] },
{ refine le_antisymm norm_mk_pi_algebra_fin_succ_le _,
convert ratio_le_op_norm _ (λ _, 1); [skip, apply_instance],
simp }
end
end
variables (𝕜 ι)
/-- The canonical continuous multilinear map on `𝕜^ι`, associating to `m` the product of all the
`m i` (multiplied by a fixed reference element `z` in the target module) -/
protected def mk_pi_field (z : G) : continuous_multilinear_map 𝕜 (λ(i : ι), 𝕜) G :=
multilinear_map.mk_continuous
(multilinear_map.mk_pi_ring 𝕜 ι z) (‖z‖)
(λ m, by simp only [multilinear_map.mk_pi_ring_apply, norm_smul, norm_prod,
mul_comm])
variables {𝕜 ι}
@[simp] lemma mk_pi_field_apply (z : G) (m : ι → 𝕜) :
(continuous_multilinear_map.mk_pi_field 𝕜 ι z : (ι → 𝕜) → G) m = (∏ i, m i) • z := rfl
lemma mk_pi_field_apply_one_eq_self (f : continuous_multilinear_map 𝕜 (λ(i : ι), 𝕜) G) :
continuous_multilinear_map.mk_pi_field 𝕜 ι (f (λi, 1)) = f :=
to_multilinear_map_injective f.to_multilinear_map.mk_pi_ring_apply_one_eq_self
@[simp] lemma norm_mk_pi_field (z : G) : ‖continuous_multilinear_map.mk_pi_field 𝕜 ι z‖ = ‖z‖ :=
(multilinear_map.mk_continuous_norm_le _ (norm_nonneg z) _).antisymm $
by simpa using (continuous_multilinear_map.mk_pi_field 𝕜 ι z).le_op_norm (λ _, 1)
lemma mk_pi_field_eq_iff {z₁ z₂ : G} :
continuous_multilinear_map.mk_pi_field 𝕜 ι z₁ = continuous_multilinear_map.mk_pi_field 𝕜 ι z₂ ↔
z₁ = z₂ :=
begin
rw [← to_multilinear_map_injective.eq_iff],
exact multilinear_map.mk_pi_ring_eq_iff
end
lemma mk_pi_field_zero :
continuous_multilinear_map.mk_pi_field 𝕜 ι (0 : G) = 0 :=
by ext; rw [mk_pi_field_apply, smul_zero, continuous_multilinear_map.zero_apply]
lemma mk_pi_field_eq_zero_iff (z : G) :
continuous_multilinear_map.mk_pi_field 𝕜 ι z = 0 ↔ z = 0 :=
by rw [← mk_pi_field_zero, mk_pi_field_eq_iff]
variables (𝕜 ι G)
/-- Continuous multilinear maps on `𝕜^n` with values in `G` are in bijection with `G`, as such a
continuous multilinear map is completely determined by its value on the constant vector made of
ones. We register this bijection as a linear isometry in
`continuous_multilinear_map.pi_field_equiv`. -/
protected def pi_field_equiv : G ≃ₗᵢ[𝕜] (continuous_multilinear_map 𝕜 (λ(i : ι), 𝕜) G) :=
{ to_fun := λ z, continuous_multilinear_map.mk_pi_field 𝕜 ι z,
inv_fun := λ f, f (λi, 1),
map_add' := λ z z', by { ext m, simp [smul_add] },
map_smul' := λ c z, by { ext m, simp [smul_smul, mul_comm] },
left_inv := λ z, by simp,
right_inv := λ f, f.mk_pi_field_apply_one_eq_self,
norm_map' := norm_mk_pi_field }
end continuous_multilinear_map
namespace continuous_linear_map
lemma norm_comp_continuous_multilinear_map_le (g : G →L[𝕜] G')
(f : continuous_multilinear_map 𝕜 E G) :
‖g.comp_continuous_multilinear_map f‖ ≤ ‖g‖ * ‖f‖ :=
continuous_multilinear_map.op_norm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) $ λ m,
calc ‖g (f m)‖ ≤ ‖g‖ * (‖f‖ * ∏ i, ‖m i‖) : g.le_op_norm_of_le $ f.le_op_norm _
... = _ : (mul_assoc _ _ _).symm
variables (𝕜 E G G')
/-- `continuous_linear_map.comp_continuous_multilinear_map` as a bundled continuous bilinear map. -/
def comp_continuous_multilinear_mapL :
(G →L[𝕜] G') →L[𝕜] continuous_multilinear_map 𝕜 E G →L[𝕜] continuous_multilinear_map 𝕜 E G' :=
linear_map.mk_continuous₂
(linear_map.mk₂ 𝕜 comp_continuous_multilinear_map (λ f₁ f₂ g, rfl) (λ c f g, rfl)
(λ f g₁ g₂, by { ext1, apply f.map_add }) (λ c f g, by { ext1, simp }))
1 $ λ f g, by { rw one_mul, exact f.norm_comp_continuous_multilinear_map_le g }
variables {𝕜 G G'}
/-- `continuous_linear_map.comp_continuous_multilinear_map` as a bundled
continuous linear equiv. -/
def _root_.continuous_linear_equiv.comp_continuous_multilinear_mapL (g : G ≃L[𝕜] G') :
continuous_multilinear_map 𝕜 E G ≃L[𝕜] continuous_multilinear_map 𝕜 E G' :=
{ inv_fun := comp_continuous_multilinear_mapL 𝕜 _ _ _ g.symm.to_continuous_linear_map,
left_inv := begin
assume f,
ext1 m,
simp only [comp_continuous_multilinear_mapL, continuous_linear_equiv.coe_def_rev,
to_linear_map_eq_coe, linear_map.to_fun_eq_coe, coe_coe, linear_map.mk_continuous₂_apply,
linear_map.mk₂_apply, comp_continuous_multilinear_map_coe, continuous_linear_equiv.coe_coe,
function.comp_app, continuous_linear_equiv.symm_apply_apply],
end,
right_inv := begin
assume f,
ext1 m,
simp only [comp_continuous_multilinear_mapL, continuous_linear_equiv.coe_def_rev,
to_linear_map_eq_coe, linear_map.mk_continuous₂_apply, linear_map.mk₂_apply,
linear_map.to_fun_eq_coe, coe_coe, comp_continuous_multilinear_map_coe,
continuous_linear_equiv.coe_coe, function.comp_app, continuous_linear_equiv.apply_symm_apply],
end,
continuous_to_fun :=
(comp_continuous_multilinear_mapL 𝕜 _ _ _ g.to_continuous_linear_map).continuous,
continuous_inv_fun :=
(comp_continuous_multilinear_mapL 𝕜 _ _ _ g.symm.to_continuous_linear_map).continuous,
.. comp_continuous_multilinear_mapL 𝕜 _ _ _ g.to_continuous_linear_map }
@[simp] lemma _root_.continuous_linear_equiv.comp_continuous_multilinear_mapL_symm
(g : G ≃L[𝕜] G') :
(g.comp_continuous_multilinear_mapL E).symm = g.symm.comp_continuous_multilinear_mapL E := rfl
variables {E}
@[simp] lemma _root_.continuous_linear_equiv.comp_continuous_multilinear_mapL_apply
(g : G ≃L[𝕜] G') (f : continuous_multilinear_map 𝕜 E G) :
g.comp_continuous_multilinear_mapL E f = (g : G →L[𝕜] G').comp_continuous_multilinear_map f :=
rfl
/-- Flip arguments in `f : G →L[𝕜] continuous_multilinear_map 𝕜 E G'` to get
`continuous_multilinear_map 𝕜 E (G →L[𝕜] G')` -/
def flip_multilinear (f : G →L[𝕜] continuous_multilinear_map 𝕜 E G') :
continuous_multilinear_map 𝕜 E (G →L[𝕜] G') :=
multilinear_map.mk_continuous
{ to_fun := λ m, linear_map.mk_continuous
{ to_fun := λ x, f x m,
map_add' := λ x y, by simp only [map_add, continuous_multilinear_map.add_apply],
map_smul' := λ c x, by simp only [continuous_multilinear_map.smul_apply, map_smul,
ring_hom.id_apply] }
(‖f‖ * ∏ i, ‖m i‖) $ λ x,
by { rw mul_right_comm, exact (f x).le_of_op_norm_le _ (f.le_op_norm x) },
map_add' := λ _ m i x y,
by { ext1, simp only [add_apply, continuous_multilinear_map.map_add, linear_map.coe_mk,
linear_map.mk_continuous_apply]},
map_smul' := λ _ m i c x,
by { ext1, simp only [coe_smul', continuous_multilinear_map.map_smul, linear_map.coe_mk,
linear_map.mk_continuous_apply, pi.smul_apply]} }
‖f‖ $ λ m,
linear_map.mk_continuous_norm_le _
(mul_nonneg (norm_nonneg f) (prod_nonneg $ λ i hi, norm_nonneg (m i))) _
end continuous_linear_map
lemma linear_isometry.norm_comp_continuous_multilinear_map
(g : G →ₗᵢ[𝕜] G') (f : continuous_multilinear_map 𝕜 E G) :
‖g.to_continuous_linear_map.comp_continuous_multilinear_map f‖ = ‖f‖ :=
by simp only [continuous_linear_map.comp_continuous_multilinear_map_coe,
linear_isometry.coe_to_continuous_linear_map, linear_isometry.norm_map,
continuous_multilinear_map.norm_def]
open continuous_multilinear_map
namespace multilinear_map
/-- Given a map `f : G →ₗ[𝕜] multilinear_map 𝕜 E G'` and an estimate
`H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖`, construct a continuous linear
map from `G` to `continuous_multilinear_map 𝕜 E G'`.
In order to lift, e.g., a map `f : (multilinear_map 𝕜 E G) →ₗ[𝕜] multilinear_map 𝕜 E' G'`
to a map `(continuous_multilinear_map 𝕜 E G) →L[𝕜] continuous_multilinear_map 𝕜 E' G'`,
one can apply this construction to `f.comp continuous_multilinear_map.to_multilinear_map_linear`
which is a linear map from `continuous_multilinear_map 𝕜 E G` to `multilinear_map 𝕜 E' G'`. -/
def mk_continuous_linear (f : G →ₗ[𝕜] multilinear_map 𝕜 E G') (C : ℝ)
(H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) :
G →L[𝕜] continuous_multilinear_map 𝕜 E G' :=
linear_map.mk_continuous
{ to_fun := λ x, (f x).mk_continuous (C * ‖x‖) $ H x,
map_add' := λ x y, by { ext1, simp only [_root_.map_add], refl },
map_smul' := λ c x, by { ext1, simp only [smul_hom_class.map_smul], refl } }
(max C 0) $ λ x, ((f x).mk_continuous_norm_le' _).trans_eq $
by rw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul]
lemma mk_continuous_linear_norm_le' (f : G →ₗ[𝕜] multilinear_map 𝕜 E G') (C : ℝ)
(H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) :
‖mk_continuous_linear f C H‖ ≤ max C 0 :=
begin
dunfold mk_continuous_linear,
exact linear_map.mk_continuous_norm_le _ (le_max_right _ _) _
end
lemma mk_continuous_linear_norm_le (f : G →ₗ[𝕜] multilinear_map 𝕜 E G') {C : ℝ} (hC : 0 ≤ C)
(H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) :
‖mk_continuous_linear f C H‖ ≤ C :=
(mk_continuous_linear_norm_le' f C H).trans_eq (max_eq_left hC)
/-- Given a map `f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G)` and an estimate
`H : ∀ m m', ‖f m m'‖ ≤ C * ∏ i, ‖m i‖ * ∏ i, ‖m' i‖`, upgrade all `multilinear_map`s in the type to
`continuous_multilinear_map`s. -/
def mk_continuous_multilinear (f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G)) (C : ℝ)
(H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ C * (∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) :
continuous_multilinear_map 𝕜 E (continuous_multilinear_map 𝕜 E' G) :=
mk_continuous
{ to_fun := λ m, mk_continuous (f m) (C * ∏ i, ‖m i‖) $ H m,
map_add' := λ _ m i x y, by { ext1, simp },
map_smul' := λ _ m i c x, by { ext1, simp } }
(max C 0) $ λ m, ((f m).mk_continuous_norm_le' _).trans_eq $
by { rw [max_mul_of_nonneg, zero_mul], exact prod_nonneg (λ _ _, norm_nonneg _) }
@[simp] lemma mk_continuous_multilinear_apply (f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G))
{C : ℝ} (H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ C * (∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) (m : Π i, E i) :