fa10_sub_eng.srt

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Hello and welcome back to

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functional analysis.

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And first many thanks to

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all the nice people that

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support me on Steady or

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paypal.

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And I can also tell you that

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now you find the PDF

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versions of all new videos

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in the description below.

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So this is part 10 today

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and we will still talk about

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inner products.

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In particular, I want to

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prove the Cauchy Schwarz

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inequality

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depending where you come

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from.

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The equality is known by

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different names.

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But I stay here with the

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most common one.

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And here it is named after the

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French mathematician Cauchy

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and a German mathematician

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Schwarz.

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Indeed, this inequality is

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very important because it

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holds for all inner products.

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Therefore, let's choose an

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F vector space X and an

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inner product.

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And we also consider the

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corresponding norm which

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is the square root of the

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inner product.

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And then with these notations,

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the following inequality

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holds, we

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look at the inner product

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X with Y.

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And because it could be in

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general a complex number,

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we need the absolute value

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here.

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And then we can say this

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is less or equal than the

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two norms multiplied.

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And that's the whole Cauchy

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Schwarz inequality.

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Now we can visualize it with

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the same picture we had when

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we started with the inner

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products in this

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multiplication X with

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Y only the parallel

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part of Y, the yellow part

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here should matter.

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Then this picture tells you,

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yeah, of course, the length

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of the yellow part is less

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or equal than the length

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of the wet arrow here.

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With this in mind, we also

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get a result in which cases

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the equality here holds

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indeed, this should only

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be the case when the arrows

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go in the same direction.

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In other words, X and Y

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are linearly dependent

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vectors.

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OK.

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Now the picture gets us in

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the correct direction, but

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we don't have any choice.

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We need to prove the inequality.

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Now.

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OK.

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So let's start with an easy

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case, let's call it the first

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case where X is the zero

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vector.

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Of course, there we know

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the left hand side has to

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be zero

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simply by the linearity,

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we can just pull out the

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factor zero.

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And of course, the right

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hand side is also zero

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because the norm of X has

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to be zero here.

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In particular, the general

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inequality is

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obviously fulfilled for this

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simple case.

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Now you might already guess

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the actual interesting case

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would be the second one

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here X is not a zero

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vector, which means we can

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divide by the norm of X.

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And that's what we do immediately,

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I want to define X hat

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as the normalized vector

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X, this means

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that X hat has length one.

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So it just gives the direction

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of the vector X and

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then we calculate the inner

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product with X hat and

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Y and go into the

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direction of X hat.

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Now, you might recognize

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that because if we have

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considered the normal Euclidean

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geometry in the plane, this

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picture would be completely

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correct.

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And this expression is

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exactly this orange line

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by our abstraction.

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This should be the same here.

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So let's call the orange arrow

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just y

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parallel in the

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Euclidean geometry.

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This is known as the orthogonal

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projection of Y on to

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X.

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So it makes sense to do the

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same for a general in our

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product.

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Now the gray line here is

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the orthogonal part which

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we also can calculate.

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Now this is

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simply given by Y

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minus the parallel part.

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OK.

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So we defined some vectors

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we want to deal with.

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But now we need an idea how

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to get to the inequality.

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Indeed, the whole idea is

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that we can calculate the

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length or the norm of

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this or formal part

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simply because we know the

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norm can't be negative.

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Now, on the line we can put

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in the definition and then

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of course, also the definition

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of Y parallel.

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Now if you see something

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like this calculating norms,

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but you have an inner product,

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it's better to use squares.

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So square everything and

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the square roots will vanish.

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And now you can see on the

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right hand side, we have

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this long vector in the middle,

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in both components of the

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inner product.

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So it looks like this.

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And in order to simplify

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this, we will use the

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linearity.

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And this linearity in the

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second component means we

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can pull out this minus

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sign here.

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So we have here a product

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with Y minus the other

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part.

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In the next step, we want

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to pull out the minus sign

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here And here, and we

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know we can do that because

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the inner product is

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conjugate linear in the

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first argument.

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So minus signs are not a

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problem.

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And scalars get a complex

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conjugation.

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So that's the first part

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and the second part would

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be this vector with y?

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And of course, now we do

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the same with the second

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inner product here.

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OK.

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Here we have Y with the vector

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on the right hand side.

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And now we have minus minus.

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So plus the rest.

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In fact, here we have the

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same vector left and right

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in the inner product.

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Therefore, now comes the

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part where we can rewrite

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everything.

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Again.

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The first thing is the norm

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of Y squared.

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For the two parts in the

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middle, we see they are almost

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the same, the one is the

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complex conjugate of the

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other one.

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So we can write it in this

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way with parentheses, we

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put a bar over the second

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part.

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And finally, the last part

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is the norm of this vector

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squared.

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In the next equality here,

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I want to simplify that even

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more because you see

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here's a complex number plus

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the complex conjugate of

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this number, which means

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this is two times the real

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part of this complex

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number, which means that

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we can write it like this.

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And now in the third part,

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we can pull out the scalar

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which means it comes out

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with the absolute value squared,

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you see it remains the norm

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of x hat where we already

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know this is one.

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Now, finally, I want to simplify

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the middle part here, which

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to be honest, we could have

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done before.

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But then we would have done

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it two times.

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I want to show you visually

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what we want to do.

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We pull this scalar out

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from the first argument of

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the inner product, but

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then it gets a complex

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conjugation.

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So let's fill in the gaps

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again.

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And then you see we have

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the same complex number left

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and right.

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But the one is the complex

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conjugate of the other one.

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In other words, it's the

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absolute value of the complex

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number squared.

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And since that is clearly

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a real number, we can omit

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the real part here.

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So in summary, we have the

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norm of Y squared minus two

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times this number,

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but also plus one

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times the same number.

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Therefore, we can put both

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things together and have

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a simple result here,

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the norm of Y squared minus

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the inner product in absolute

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value squared.

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So this is our right hand

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side and on the left, we

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have the zero.

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Hence, in the next step,

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I want to bring the inner

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product to the other side.

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Now, you can see we are almost

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there with our inequality.

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The only thing missing is

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that we have to translate

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back X hat

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which is not so hard to do

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because we define it as X

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divided by the norm of X.

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Of course, we do the same

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as always, we pull out one

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divided by the norm of X.

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Hence, that's what we get.

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And you see, we just have

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to multiply with the norm

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of X squared on both sides.

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And then taking the square

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root on both sides, we get

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out our Cauchy Schwarz

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inequality.

300
00:07:49,549 --> 00:07:51,130
Well, that's the whole proof

301
00:07:51,140 --> 00:07:52,279
of the inequality.

302
00:07:52,510 --> 00:07:54,170
I will skip the part about

303
00:07:54,179 --> 00:07:55,970
the equality now because

304
00:07:55,980 --> 00:07:57,790
I want to use time to show

305
00:07:57,799 --> 00:07:58,950
you that the symbol we use

306
00:07:58,959 --> 00:08:00,600
here is indeed the norm.

307
00:08:00,609 --> 00:08:02,059
So it fulfills the triangle

308
00:08:02,070 --> 00:08:03,029
inequality.

309
00:08:03,190 --> 00:08:04,700
And we will do that by

310
00:08:04,709 --> 00:08:06,220
using the Cauchy Schwarz

311
00:08:06,230 --> 00:08:07,059
inequality.

312
00:08:07,920 --> 00:08:09,880
In fact, the triangle inequality

313
00:08:09,890 --> 00:08:11,200
is most of the time, the

314
00:08:11,209 --> 00:08:12,820
hardest part of a proof that

315
00:08:12,829 --> 00:08:13,989
something is a norm.

316
00:08:15,179 --> 00:08:16,779
So let's do that very quickly.

317
00:08:16,790 --> 00:08:18,059
Of course, we square the

318
00:08:18,070 --> 00:08:18,839
norm again

319
00:08:19,820 --> 00:08:21,119
because then we don't need

320
00:08:21,130 --> 00:08:22,369
the square root, we can just

321
00:08:22,380 --> 00:08:24,350
write X plus YX

322
00:08:24,359 --> 00:08:26,000
plus Y in the inner product.

323
00:08:26,420 --> 00:08:28,019
The linearity now gets us

324
00:08:28,029 --> 00:08:29,920
to a similar result as before

325
00:08:30,809 --> 00:08:32,750
namely the norm of X and

326
00:08:32,760 --> 00:08:33,619
Y squared.

327
00:08:33,630 --> 00:08:34,799
And in the middle, we get

328
00:08:34,808 --> 00:08:36,479
two times the real part of

329
00:08:36,489 --> 00:08:37,239
the product.

330
00:08:37,890 --> 00:08:39,210
Now, of course, if we use

331
00:08:39,219 --> 00:08:40,409
the absolute value instead

332
00:08:40,419 --> 00:08:42,020
of the real part, it gets

333
00:08:42,030 --> 00:08:42,489
bigger.

334
00:08:43,369 --> 00:08:44,710
And of course, here we now

335
00:08:44,719 --> 00:08:46,599
can use our Cauchy Schwarz

336
00:08:46,609 --> 00:08:47,510
inequality.

337
00:08:48,080 --> 00:08:49,630
And now knowing the binomial

338
00:08:49,640 --> 00:08:51,109
theorem, you see this is

339
00:08:51,119 --> 00:08:52,669
just the square of the sum

340
00:08:52,679 --> 00:08:53,729
of the two norms

341
00:08:54,320 --> 00:08:55,419
taking the square root on

342
00:08:55,429 --> 00:08:56,109
both sides.

343
00:08:56,119 --> 00:08:57,640
You see this is simply the

344
00:08:57,650 --> 00:08:59,130
triangle inequality for the

345
00:08:59,140 --> 00:09:00,820
norm we defined by the inner

346
00:09:00,830 --> 00:09:01,349
product.

347
00:09:02,169 --> 00:09:03,210
The other two properties

348
00:09:03,219 --> 00:09:04,609
for norm shouldn't be a problem

349
00:09:04,619 --> 00:09:05,250
to show.

350
00:09:05,299 --> 00:09:06,789
And therefore now it makes

351
00:09:06,799 --> 00:09:08,650
sense to use this symbol

352
00:09:08,659 --> 00:09:09,770
and call it a norm in an

353
00:09:09,780 --> 00:09:11,229
inner product space.

354
00:09:11,770 --> 00:09:12,229
OK?

355
00:09:12,239 --> 00:09:13,190
I think that's good enough

356
00:09:13,200 --> 00:09:13,909
for today.

357
00:09:13,929 --> 00:09:15,169
In the next videos, we will

358
00:09:15,179 --> 00:09:16,590
talk a little bit more about

359
00:09:16,599 --> 00:09:17,669
the geometry.

360
00:09:17,679 --> 00:09:18,989
An inner product gives the

361
00:09:19,000 --> 00:09:20,780
space and I

362
00:09:20,789 --> 00:09:22,280
already told you if you need

363
00:09:22,289 --> 00:09:23,650
it, you can download the

364
00:09:23,659 --> 00:09:25,169
PDF version of this video

365
00:09:25,179 --> 00:09:27,159
in the description and of

366
00:09:27,169 --> 00:09:28,679
course use the comments if

367
00:09:28,690 --> 00:09:29,599
you have questions.

368
00:09:29,719 --> 00:09:30,979
Otherwise I see you in the

369
00:09:30,989 --> 00:09:31,609
next video.

370
00:09:32,590 --> 00:09:33,609
Have a nice day.

371
00:09:33,700 --> 00:09:34,289
Bye.