Show that $T$ is a bounded operator I have the following problem:
On $C^1([0,1],\mathbb{R})$ consider the norm $\Vert f \Vert=\Vert f'\Vert_\infty+\Vert f \Vert_\infty$.
Let $g,h\in C([0,1],\mathbb{R})$ be fixed and $C([0,1],\mathbb{R})$ equipped with the $\Vert \cdot \Vert_\infty $-norm. Define the operator $T:C^1([0,1],\mathbb{R})\rightarrow C([0,1],\mathbb{R})\\
(Tf)(t)=g(t)f'(t)+h(t)f(t).$
Show that $T$ is linear and bounded.
I have proved that $T$ is linear, but I´m having some problems with showing that $T$ is bounded.
This it what I´ve tried:$\Vert (Tf)(t) \Vert = \Vert g(t)f'(t)+h(t)f(t) \Vert = \Vert [g(t)f'(t)+h(t)f(t)]'\Vert_\infty+\Vert g(t)f'(t)+h(t)f(t) \Vert_\infty = \Vert g'(t)f'(t)+g(t)f''(t)+h'(t)f(t)+h(t)f'(t)\Vert_\infty+\Vert g(t)f'(t)+ h(t)f(t) \Vert_\infty \leq \Vert g'(t)f'(t)\Vert_\infty+\Vert g(t)f''(t)\Vert_\infty +\Vert h'(t)f(t)\Vert_\infty+\Vert h(t)f'(t)\Vert_\infty+\Vert g(t)f'(t)\Vert_\infty+\Vert h(t)f(t) \Vert_\infty $
I'm trying to prove $\Vert (Tf)(t) \Vert \leq \mu \Vert f(t)  \Vert$  for some $\mu>0$ according to the definition of a bounded linear operator. I don't see how what I've done so far will lead to this; am I doing something wrong?
 A: The co-domain is equipped with the sup norm so you only have to look at $\|gf'+hf\|_{\infty}$ and not $\|Tf\|$. Once you realize this I am sure you can prove boundeness of $T$ easily. In fact $\|T\|\leq \max \{\|g\|_{\infty},\|h\|_{\infty}\}$
A: Let me abstract some of this away so you can see more clearly what is going on here. Let $X = C^1([0,1])$ and $Y = C([0,1])$. $X$ is equipped with norm $\|f\|_X = \|f\|_{\infty} + \|f'\|_{\infty}$. $Y$ is equipped with norm $\|f\|_Y = \|f\|_{\infty}$. $T$ is a mapping from $X$ to $Y$, so for instance the operator norm of $T$ would be
$$\|T\| = \sup_{\|f\|_X=1} \|Tf\|_Y = \sup_{\|f\|_{\infty}+\|f'\|_{\infty}=1} \|Tf\|_{\infty} = \sup_{\|f\|_{\infty}+\|f'\|_{\infty}=1} \|f'g+fh\|_{\infty}.$$
To show that $T$ is bounded, you want to show that there exists $C > 0$ such that $\|Tf\|_Y \le C \|f\|_X$, i.e.
$$\|f'g+fh\|_{\infty} \le C(\|f\|_{\infty} + \|f'\|_{\infty})$$
which is tantamount to showing $\|T\| < \infty$. It can be nontrivial to find the exact operator norm, so we tend to settle for just showing boundedness at all.
