Locally convex space which is not Banach.

Question

I know that all Banach spaces are (Hausdorff)locally convex spaces. I would like to verify that the converse is not true by giving an example of a space which is locally convex but not Banach. I am looking at the linear space $C([0,1],\left\|\cdot\right\|_1)$ of complex-valued functions defined on the compact interval $[0,1]$, where the norm is given by $$\left\|f\right\|_1=\int_{0}^{1}|f(t)|dt $$ for any $f\in C[0,1]$. The normed space $C([0,1],\left\|\cdot\right\|_1)$ is known to be not Banach. I don't have any idea if the said space is a locally convex space. Any tips? Thanks in advance...

Answer 1

This is sasisfied by the Schawrtz space, which is locally convex but isn't Banach. There is a class of spaces called Frechet spaces that are locally convex and not necessarily Banach. Wikipedia has a few examples.