Example of a Vector space

Question

I am trying to find an example of a real vector space which with respect to a norm is a Banach space but is not a Banach space with respect to another norm.

Answer 1

A typical example would be $C[0,1]$ with the infinity norm and with the one-norm.

Or $C^1[0,1]$ with $\|f\|=\|f\|_\infty+\|f'\|_\infty$ and with $\|f\|_\infty$.

Or $\ell^1(\mathbb N)$ with $\|\cdot\|_1$ and $\|\cdot\|_\infty$.

Answer 2

Let $X=C[0,1]$.

If $||f||_{\infty}= \max \{|f(x)| : x \in [0,1]\}$, then $(X, ||\cdot||_{\infty})$ is a Banach space.

If $||f||_{1}= \int_0^1 |f(x)| dx$, then $(X, ||\cdot||_{1})$ is not a Banach space.