$C^k (I, E)$ := space of $k$ times differentiable functions from an interval $I$ into a Banach space $E$.
I don't know the exact meaning of "into a Banach space". Please help me.
$C^k (I, E)$ := space of $k$ times differentiable functions from an interval $I$ into a Banach space $E$.
I don't know the exact meaning of "into a Banach space". Please help me.
Let $I \subseteq \mathbb R$ be an interval and $E$ a Banach space (i. e. a complete normed vector space over $\mathbb R$ or $\mathbb C$). A function $f\colon I \to E$ is called differentiable at $x\in I$ if the limit \[ f'(x) := \lim_{h \to 0}\frac{f(x+h) - f(x)}h \] exists. If $f$ is differentiable at all points of $I$, $f'$ is called the derivative of $f$. If $f'$ is differentiable, $f$ is called twice differentiable and so on (as allways). Now \[ C^k(I, E) = \{f \colon I \to E \mid f \text{ is $k$-times differentiable and $f^{(k)}$ is continuous} \} \]
It means each function $f$ has outputs in $E$. That is, $f:I\rightarrow E$.
If you are asking about Banach spaces themselves, they are complete, normed, linear spaces.