Proposition. Let $E_{1}$ and $E_{2}$ be normed spaces and $A: E_1 \rightarrow E_2, x \mapsto Ax$ a linear operator. Then, $A$ is continuous if and only if A is bounded.
I was able to follow the proof given by Kreyszig in his book Introductory Functional Analysis with Applications.
I was able to prove this direction by myself
bounded $\implies$ continuous. $A$ bounded $\iff$ $||Ax|| \leq ||A|| ||x||$ where $||A|| = sup \frac{|Ax|}{||x||}$ for $x \neq 0$.
My thoughts were $\forall \epsilon>0 \exists \delta >0 : ||x - x_0|| \implies ||Ax - Ax_0|| < \epsilon$. So I have to find the $\delta$. Thus I used $||Ax - Ax_0|| = ||A(x - x_0)|| \leq ||A|| ||x - x_0|| < ||A|| \delta$. And now I recalled that I want to find the $\delta$, so I looked what fits my needs and chose $\delta = \frac{\epsilon}{||A||}$ for $||A|| \neq 0$. Choosing $\epsilon$ and $x_{0}$ arbitrary shows continuity for the case $||A|| \neq 0$. Now, the case $||A|| = 0$ is trivial.
The proof by Kreyszig starts with choosing the $||A|| \neq 0$ and the $\delta = \frac{\epsilon}{||A||}$. Now, I understand that this is how we are taught to present proofs. But this does not seem like "intuitive discovery" for me. I.e. whenever I reconstruct this proof as a repetition exercise to prepare in case this question gets asked on the oral exam, I always start how I explained the proof above and not how Kreyszig starts.
Now, for the
continuous $\implies$ bounded. I tried doing this interplay of what is given and what has to be proved. But it got me only as far as unwrapping the definition of bounded and continuous. I know I have to find a $c$ such that $||Ax|| \leq c ||x||$. I got stuck for some time here and then I looked at the book. The idea/trick was to define for any $y \neq 0$
$$x = x_0 + \frac{\delta}{||y||}y.$$
From here I was able to take over and finish the proof. But the choice of choosing the above as was chosen did not occur to me. How does one "discover" this choice that is needed for the proof? Is it intuition? To me this choice seems mysterious. If someone could explain it to me would be nice.
My follow up questions are:
Since I have an oral exam upcoming that includes proving the standard theorems of the material in functional analysis, how do I remember the proofs which I cannot discover by myself using this interplay of "what is given and what is needed to be proven"? Do I simply rote memorize the key "tricks" and then rely that I can piece it together from there? Is this even an efficient method?
Also what is the purpose of such examinations where one has to basically memorize a proof or some parts of it, then present them to the examinator, pass the course and never use the proof idea ever again?
Perhaps this proof will prove more enlightening?
By continuity at the zero element, there exists $\delta>0$ such that $\|Ax\| = \|A(x-0)\|\leqslant 1$ if $\|x\|<\delta$. Then \begin{align} \|Ax\| &= \left\|\frac{\|x\|}\delta A\left(\delta\frac x{\|x\|}\right)\right\|\\ &= \frac{\|x\|}\delta\left\|A\left(\delta\frac x{\|x\|}\right)\right\|\\ &\leqslant \frac{\|x\|}\delta \cdot 1 = \frac1\delta\|x\|, \end{align} which implies that $A$ is bounded.