If $A:X \rightarrow X$ is a linear bounded operator then $e^{A}:X \rightarrow X$ is a linear bounded operator

Question

Let $L(X)$ be the space of all linear bounded operators on $X$ under the operator norm.

What I got so far was, since we know that $L(X)$ is a Banach space thus every absolutely convergent series is convergent in $L(X)$ thus all we need to prove is that $\sum_{n=0}^{\infty}\| \frac{A^n}{n!} \|$ converges in $\mathbb{R}$

Answer 1

That's the operator norm that you're working with. Therefore, $(\forall n\in\mathbb{N}):\lVert A^n\rVert\leqslant\lVert A\rVert^n$. So, by the ratio test and the comparison test, your series converges for each $A$.

Answer 2

It is routine to check that $A$ is linear on its domain. You need only show that it is bounded, and this follows from the fact that $\|e^A\|\leq e^{\|A\|}$, which is seen in the image you provided.