A is bounded implies A is continuous?

Question

I have two questions which had me struggled for hours:

Note that the definition suggests only a "IF" condition.

What happens if A is bounded?

Secondly, how can I show that If A is bounded then A is continuous at 0?

My notes suggests:

I don't quite appreciate the way the proof is set out as it is. In particular, what is the motivation for assuming ||v|| >$\epsilon$?

Thanks in advance.

Answer 1

The answer to your first question is that if $A$ is bounded then there exists a constant $k$ such that for all $v \in V$ one have $\Vert Av \Vert \leq k \Vert v \Vert$. It's a definition, not a theorem (one could, in the definition, write "if and only if".)

For the second question, you wish to show that given $A$ bounded and any $\epsilon >0$ there exists $\delta > 0$ such that whenever $\Vert v \Vert < \delta$, $\Vert Av - A0 \Vert = \Vert Av \Vert < \epsilon$. Let $\delta = \epsilon/k$ where $k$ is the constant bounding $A$. Then we have $$ \Vert Av \Vert \leq k\Vert v \Vert < \epsilon, $$ provided $\Vert v \Vert < \delta$.