We define a map $T: L^2[0,1] \to L^2[0,1]$ s.t $T(f)=\int_0^1 fgdx$. Determine $||T||$.

Question

Let $g \in L^ {\infty}$

we define a map $T: L^2[0,1] \to L^2[0,1]$ s.t $T(f)=\int_0^1 fgdx$. Determine $||T||$.

I have seen that $||T||=\sup \{||T(f)|| : ||f||=1\} \leq ||g||_{\infty}$

How to prove the converse? I also know that the set $E_n=\{x \in [0,1]: |g(x) >||g||_{\infty} - \frac1n\}$ has positive measure.

So please check what I have done is right or not and provide the answer in details.

Answer 1

I hope it helps. (................)

Answer 2

HINT:

Note that $g$ also lies in $L_{2}([0,1])$. By the Cauchy-Schwarz we have ...