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Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg:

Let $C[0,1]$ denote the set of all (real- or complex-valued) functions defined and continuous on the closed interval $[0,1]$ with the norm defined as follows: $$\Vert x \Vert_{C[0,1]} \colon= \max_{t \in [0,1]} \vert x(t) \vert \, \, \, \forall x \in C[0,1]. $$

Let $T \colon C[0,1] \to C[0,1]$ be defined by $$Tx \colon= \int_{0}^t x(\tau) \, d \tau \, \, \, \forall x \in C[0,1].$$

What is the range of $T$?

Is $T$ injective? And if so, then what is the inverse $T^{-1}$? Is $T^{-1}$ bounded?

I know that $T$ is bounded, for, given any $x \in C[0,1]$, we have $$ \Vert Tx \Vert_{C[0,1]} = \max_{t\in[0,1]} \vert \int_0^t x(\tau) \, d \tau \vert \leq \max_{t\in[0,1]} (\int_0^t \vert x(\tau) \vert \, d \tau ) = \int_0^1 \vert x(\tau) \vert \, d \tau \leq \int_0^1 \max_{\tau\in[0,1]} \vert x(\tau) \vert \, d \tau = \int_0^1 \Vert x \Vert_{C[0,1]} \, d \tau = \Vert x \Vert_{C[0,1]}.$$

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    $\begingroup$ Here's something to get you started: by the "fundamental theorem of calculus", the range of $T$ consists of all functions $f\in C[0,1]$ such that $f$ is continuously differentiable in $(0,1),$ $f$ is continuously differentiable from the right in $0,$ $f$ is continuously differentiable from the left in $1,$ and $f(0) = 0.$ $\endgroup$ – jflipp Dec 18 '14 at 9:09
  • $\begingroup$ Can you please also give a detailed proof of what you've stated in your comment? I'm a bit rusty on this stuff, I must admit, although I did last year cover chapters 1 throuhg 8 in Apostol's Calculus volume 1 fully and even did all the exercises! $\endgroup$ – Saaqib Mahmood Dec 22 '14 at 5:09
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Let $g$ be an element of the range of $T$. Then there exists $f\in C[0,1]$ such that $g=Tf$. As jflipp commented, this implies that for $t\in[0,1]$, $g'(t)=\frac{d}{dt}\int_0^t f(\tau)\,d\tau=f(t)$; this is one version of the Fundamental Theorem of Calculus you may look up in your Apostol text. Thus $g$ is continuously differentiable on $[0,1]$ (take care of the special meaning this has at the endpoints), and $g(0)=0$.

Conversely, suppose that $g$ is continuously differentiable on $[0,1]$ and $g(0)=0$. Then for $t\in[0,1]$, $\int_0^t g'(\tau)\,d\tau=g(t)-g(0)=g(t)$, by another version of the Fundamental Theorem of Calculus you may look up in your Apostol text. Thus $g=Tg'$ is in the range of $T$.

Thus the range is determined. To show that $T$ is injective, suppose that $Tf=0$. Then $0=(Tf)'=f$.

Because $T$ is injective, we can define $T^{-1}: T(C[0,1])\to C[0,1]$, which is determined by the identity $T(T^{-1}g)=g$, where $g$ is continuously differentiable and $g(0)=0$. As observed above, for this to be true requires $g'=T^{-1}g$, and sure enough, defining $T^{-1}g = g'$ yields $T(T^{-1}g))=Tg' = g$. Unboundedness of $T^{-1}$ can be seen by considering $T^{-1}f_n$ for $f_n(t)=t^n$.

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