# Frechet Space vs Banach Space

What is Frechet Space? Is a Banach a Frechet space? If not, why? If yes, how do we prove it? According to Wikipedia,

Frechet spaces are locally convex topological space that is complete with respect to translation-invariant metric.
It also says that:
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm).
So I am guessing that Frechet spaces are Banach Spaces. But how do we prove it?

• $X$ is a generalization of $Y$ meaning $Y\subseteq X$... (i.e. every Banach space is a Frechet space) – YuiTo Cheng May 23 '19 at 10:54
• Oh, so I got it wrong. Thanks. – Octagonal Monk May 23 '19 at 11:00

Let $$(B, || \cdot||)$$ be a Banach space. Then $$d(x,y):=||x-y||$$ is a translation-invariant metric on $$B$$ and $$B$$ is complete with respect to this metric.

A basis of neighborhoods of $$0$$ is given by the sets $$B_{\epsilon}:=\{x \in B:||x||< \epsilon\},$$ where $$\epsilon >0$$.

Each $$B_{\epsilon}$$ is convex.

Conclusion: a Banach is a Frechet space.

Fréchet spaces are more general than Banach spaces; every Banach space is a Fréchet space but not vice-versa. If I recall correctly, the difference is that where Banach spaces have a norm inducing the metric in which they are complete, a Fréchet space need not have a norm, just a metric. Although your quote says about the same.