I'm a bit stuck on a exercise, here it is. Let $X = C[0,1]$ be the space of continous functions on $[0,1]$ with real values.
X is a normed space with respect to the $L²$ norm: $(\int_{0}^{1} |f(t)| dt)^\frac{1}{2}$.
Let T be a linear operator defined on X, such that:
$ For f \in X, $ $$Tf = \int _{0}^{1} x²f(x) dx$$
The first question is to show that T is bounded for all $f$ in $X$, which I've completed thanks to Cauchy Schwarz and this is fine because $x² \in X$ like $f(x)$, (hence $\in L²$), and therefore I've found:
$$||T|| \le \frac{1}{\sqrt 5}$$
Then we're asked to compute (or derivate ?) the operator norm of $T$, that is to show that $||T|| = \frac{1}{\sqrt 5}$.
I can't find any form of $f$ in $X$ such that $||f||_{2} = 1 $, and I end up having $||T|| \ge \frac{1}{\sqrt{5}}$
Can anyone give me a clue on how to prove this ?
The function $g(x) = \sqrt{5}x^2$ has the desired properties: $\vert\vert g \vert\vert = 1$ and $Tg = 1/\sqrt{5}$.
How did I find $g$? The Cauchy-Schwarz-inequality is an equality, if and only if the two functions you're estimating are linearly dependent. Thus it makes sense to consider functions $ax^2$.