I need a reference to the following (known?)



> 
> **Fact.** Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of linear metric spaces.
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This seems to be a basic fact in the theory of topological vector spaces, but I did not find it in the classical textbooks: Bourbaki, Schaefer, Rolewicz, Robertson&Robertson. If you know a suitable reference, please help!



Every linear topological space embeds into the Tychonoff product of linear metric spaces
In Jarchow's *Locally Convex Spaces* Theorems [2.7.3](https://books.google.com/books?id=5tYACAAAQBAJ&pg=PA39) and [2.9.2](https://books.google.com/books?id=5tYACAAAQBAJ&pg=PA43) on pages 39 and 43 together say that a *Hausdorff* topological vector space is linearly homeomorphic to a dense subspace of a projective limit of a projective system of metrizable topological vector spaces. Noting that in section 2.6 ([p. 37](https://books.google.com/books?id=5tYACAAAQBAJ&pg=PA37)) Jarchow defines *the* projective limit as a certain subspace of the product, this contains the required assertion *provided* that the space is Hausdorff.


