I'm currently studying about the normed space of linear continous mapping between normed spaces. Here is some notation I will be using:
Let $\mathscr{L}(X,Y)$ be the normed space of continous linear mappings from $X$ to $Y$ with the norm $||A|| = \sup\limits_{x \in X \setminus \{\theta\}} \dfrac{||A(x)||_Y}{||x||_X}$, where $||.||_Y$ and $||.||_X$ are norms of the spaces $Y$ and $X$, respectively.
The exponential mapping of a linear mapping from a normed space $X$ to itself is defined as following:
Let $A \in \mathscr{L}(X,X) = \mathscr{L}(X)$, then the series $\sum\limits_{n=0}^\infty \dfrac{A^n}{n!}$ coverages in $\mathscr{L} (X)$ (proved). We denine the mapping $e^A$ to be the value of the series.
With all of the above notations, here is my problem:
Prove that $e^A = \lim\limits_{n \to \infty} \left(I + \dfrac{A}{n} \right)^n$, where $I$ is the identity mapping from $X$ to $X$ and the notation $B^n$ means $B \circ B \circ ... \circ B$ ($n$ times).
In the progress of finding a solution to that problem, I have thought about the possibility that $A \circ (B + C) = A \circ B + A \circ C$, however I cannot prove or disprove my assumption.
Please tell me if I am following a correct direction. Because, if such thing happens to be true, I could use a similar technique to the proof that $e^x = \lim\limits_{n\to \infty} \left(1 + \dfrac{x}{n} \right)^n$.
All help is greatly appriciated. Thank you.
Yes, $A\circ(B+C)=A\circ B+A\circ C$, since both operators map each vector $v$ into $A\bigl(B(v)\bigr)+A\bigl(C(v)\bigr)$.
And, yes, you can prove what you wish to prove basically by the same method of proving that $\lim_{n\to\infty}\left(1+\frac xn\right)^n=e^x$.