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You have probably seen somewhere that all separable Frechet spaces are homeomorphic. For uniform homeomorphism this isn't true. The first result stated in lecture notes of Pisier (ask Google for uniformly homeomorphic Banach spaces) states that a locally convex space which is uniformly homeomorphic to a Banach space is itself a Banach space (more precisely, isomorphic in the category of locally convex spaces to a Banach space).
– Jochen WengenrothNov 22 at 13:17

Thanks so much, Jochen for drawing my attention to the stated result. This result actually appears in Benjamini-Lindenstrauss "Geometric Functional Analysis" that I had seen years ago, but in a different context and so couldnt relate to it as a possible answer to my question. In particular, a nuclear Frechet sp can never be uniformly homeomorphic to a Banach space, unless both are finite dimensional, having the same dimension. However, the question for uniform homeomorphism between non-normable Frechet spaces remains!
– M A SofiDec 1 at 7:05

Actually, the stated result does not really provide a counterexample to my question. After all, the existence of a uniform homeomorphism between a nuclear Frechet space and a Banach space forces the latter space to be finite dimensional, and hence nuclear! Whether it happens in general, i.e. for non-normable Frechet spaces is the question.
– M A SofiDec 3 at 5:45

I did not claim to answer your question.
– Jochen WengenrothDec 3 at 8:22

uniformhomeomorphism this isn't true. The first result stated in lecture notes of Pisier (ask Google foruniformly homeomorphic Banach spaces) states that a locally convex space which is uniformly homeomorphic to a Banach space is itself a Banach space (more precisely, isomorphic in the category of locally convex spaces to a Banach space). – Jochen Wengenroth Nov 22 at 13:17