Frechet Space vs Banach Space

Question

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?

Answer 1

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.

Answer 2

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.