Find the operator norm of the following linear functional.

Question

Let $C[-1,1]$ be the normed linear space of all the continuous functions from $[-1,1]$ to $\mathbb F=\mathbb{R\text{ or }C}$,with the sup-norm $\|\cdot\|_\infty$ and define $T:C[-1,1]\to \mathbb F$ to be, $T(f)=\int_{-1}^0 f(t)dt-\int_0^1 f(t)dt$.I have to show that this is a bounded linear functional and find its operator norm.I have succeeded in showing that it is a bounded functional as, $|T(f)|=\int_{-1}^1 |f(t)|dt\leq \|f\|_\infty(1+1)=2\|f\|_\infty$.Now I think that the operator norm should be $2$.In order to show that,I have to exhibit a continuous function $f:[-1,1]\to \mathbb F$ such that $|T(f)|=2\|f\|_\infty$.But I am not able to find such a function.I tried to do it graphically but I think it is not easy to construct such a continuous function.I also saw a similar problem at Compute the operator norm of a linear functional. But could not understand what they are trying to do.Can someone help me?

Answer 1

Hint: Try to approximate the (discontinuous) function $$f(t) := \begin{cases} 1 & \text{if } t < 0,\\ -1 & \text{if } t \geq 0 \end{cases}$$ by continuous functions.