- #1

member 428835

## Homework Statement

Show the following space equipped with given norm is a Banach space.

Let ##C^k[a,b]## with ##a<b## finite and ##k \in \mathbb{N}## denote the set of all continuous functions ##u:[a,b]\to \mathbb R## that have continuous derivatives on ##[a,b]## to order ##k##. Define the norm $$||u||:= \sum_{j=0}^k \max_{a\leq x\leq b}|u^{(j)}(x)|$$

where ##u^{(j)}## is the ##j##th derivative.

## Homework Equations

Nothing comes to mind.

## The Attempt at a Solution

No idea how to start this. I believe a Banach space is when every Cauchy sequence converges. So then I need to show

$$||u_n - u_m||= \sum_{j=0}^k \max_{a\leq x\leq b}|u_n^{(j)}(x)-u_m^{(j)}(x)| < \epsilon$$

But how?