Isomorphisms of Fréchet Spaces

What is the proper notion of an isomorphism between Fréchet spaces? Obviously it should be a linear map. I'm just worried about the analytic structure. Should one be able to order the seminorms on each space so that the isomorphism preserves each seminorm (i.e., $q_n(\phi (v))=p_n(v)$)? Should it preserve the translation invariant metric? Or should it just be a homeomorphism?

I'm leaning towards the last one, as the other two notions seem too strong, especially the first, but I figured I'd ask here to double check before I go ahead with what I'm doing.

Thanks much!

To expand on my comment:

Fréchet spaces are a special class of topological vector spaces. Note that a topological vector space has a uniform structure coming from the underlying abelian topological group, so it makes sense to speak of completeness. A Fréchet space is a complete and metrizable locally convex topological vector space.

It is not hard to show that a continuous linear map between topological vector spaces is uniformly continuous. Thus, completeness and local convexity are both preserved under isomorphisms inside the category of topological vector spaces and continuous linear maps.

Metrizability of a locally convex topological vector space is equivalent to admitting a countable neighborhood base of $0$ consisting of convex, balanced and absorbing sets (it is often convenient to replace this base by a decreasing sequence of such sets). This property is also preserved under isomorphisms as topological vector spaces, hence in my opinion there is only one reasonable notion of isomorphism that can be considered: simply the linear homeomorphisms. By the open mapping theorem (as stated e.g. in Rudin's Functional Analysis, Theorem 2.11, page 47) a continuous linear bijection between Fréchet spaces is a homeomorphism.

As a further point, the (sequence of) semi-norms and metrics with which you can define a Fréchet space are usually not natural. That is, there are many convenient choices and it depends on the context which one is the most appropriate. This is in stark contrast to Banach spaces which come equipped with a preferred norm. As a corollary, neither preserving some (increasing) sequence seminorms nor preserving some translation invariant metric are natural in my opinion.

Added: in view of paul garret's second comment below:

One can quite easily show (as soon as the necessary language is developed) that every complete locally convex topological vector space $X$ is the (filtered projective) limit of Banach spaces (in the category of locally convex spaces).

To see this, choose a base $\{U_{\alpha}\}_{\alpha \in A}$ of the neighborhood filter of $0$, consisting of convex, balanced and absorbing sets and let $p_{\alpha}$ be Minkowski functional associated to $U_{\alpha}$. The Hausdorffification $X_{\alpha}$ of $(X, p_{\alpha})$ is easily seen to be a Banach space and because $A$ is directed by reverse inclusion so is $X_{\alpha}$. It is straightforward to check that $X = \varprojlim X_{\alpha}$ in the category of locally convex spaces. For details, I refer to Schaefer, Topological Vector Spaces, Chapter II.§5, page 51ff.

Now given that a Fréchet space admits a decreasing sequence of convex balanced and absorbing neighborhoods, it follows immediately that every Fréchet space is the projective limit of a sequence of Banach spaces. For further details see Schaefer, Chapter II.§4, page 48f (as well as Theorem I.6.1, page 28). Conversely, any limit of a sequence of Banach spaces is a Fréchet space.

In view of this, it is even less natural to expect that a (general) Fréchet space comes equipped with a preferred metric.