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?

  • 1
    $\begingroup$ $X$ is a generalization of $Y$ meaning $Y\subseteq X$... (i.e. every Banach space is a Frechet space) $\endgroup$ – YuiTo Cheng May 23 at 10:54
  • $\begingroup$ Oh, so I got it wrong. Thanks. $\endgroup$ – Octagonal Monk May 23 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.