1

For a Banach space $B$, given a function $f: \mathbb R \to B$, we can define its derivative at $x \in \mathbb R$ as $f'$ such that $$ \lim_{h\to 0} \frac{\|f(x+h) - f(x) - hf'(x)\|}{h} = 0 $$ if the limit exists.

In such case, I was wondering if it is possible to write $$ f(x+h) = f(x) + f'(x) h + o(h)? $$ Is it some generalization of Taylor expansion? How is $o(h)$ defined then? It represents a function from $\mathbb R$ to $B$, doesn't it?

Thanks and regards!

Tim
  • 47,382

1 Answers1

1

It is indeed possible to write it as an analogue of Taylor series as for $ x, h \in \mathbb{R} $ , $f'(x) $ being a continuous linear map from $\mathbb{R} $ to $B$ if $f$ is differentiable, then you have $$ f(x+h) = f(x) + f'(x)(h) + \epsilon(h) $$ where $ \epsilon(h)/|h| \rightarrow 0 $ in norm topology of $B$ as $|h|\rightarrow 0 $ in $\mathbb{R}$. Equivalently for usual definition of $o$ you can write $$ \|f(x+h)-f(x)-f'(x)(h)\| = o(|h|) $$

smiley06
  • 4,157