| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
23 |
|
$3 . \mathrm{II} . 13 \mathrm{H}$ |
2007 |
Let $V$ be the real vector space of continuous functions $f:[0,1] \rightarrow \mathbb{R}$. Show that defining
$$|f|=\int_{0}^{1}|f(x)| d x$$
makes $V$ a normed vector space.
Define $f_{n}(x)=\sin n x$ for positive integers $n$. Is the sequence $\left(f_{n}\right)$ convergent to some element of $V$ ? Is $\left(f_{n}\right)$ a Cauchy sequence in $V$ ? Justify your answers.