There may be some similar questions being asked but I didn't see anything that was quite what I was looking for so I'll ask the question here.
Let $X = (C[0,1],||.||_{\max})$ and $T:X \to X = \int^{t}_{0}x(\tau)d\tau$.
The norm is defined in the following way: for $x(t) \in C[0,1]$, $||x(t)||_{max}= \textrm{max}\{|x(t)|:t \in [0,1]\}$.
I want to show that $T$ is a bounded linear operator and find $||T||= \textrm{sup}\left\{\frac{||T(x(t))||}{||x(t)||}:x(t) \in C[0,1], x(t) \neq 0\right\}$.
Showing that $T$ is a bounded linear operator is simple but I'm not sure about finding the norm of $T$. By the Extreme Value Theorem, we know that a maximum exists for $x(t)$ and $T(x(t))$ on $[0,1]$. But the supremum of the quotient of all such values may be infinite. How do we show this is not the case?
Best Answer
When we use the bound $$\tag{*}\int_0^t|x(\tau)|\mathrm{d}\tau\leqslant \lVert x\rVert_\infty,$$ there is no loss if and only if $x$ is constant so the norm is greater than $1$.
Inequality (*) for general $x$ shows that the norm is actually $1$.