Finite products of topological vector spaces

Question

A finite product of an infinite Hilbert space is isometric to itself, since the isometry class of a Hilbert space is determined only by the cardinality of its dimension. But what about less 'nice' topological vector spaces of infinite dimension? I am in particular interested in Hausdorff topological vector spaces, Hausdorff locally convex spaces, and Banach spaces (up to isomorphism in the respective category).

Answer 1

Nice question. I think that in the category of Banach spaces and linear contractions as morphisms (i.e., $\|f(x)\|\le \|x\|$) the space $c$ of all real convergent sequences with the sup-norm is an example: The unit ball of $c$ has two extreme points (the constant sequences with value $\pm 1$) but in the product $c\times c$ (in the mentioned category, this is endowed with the maximum of the norms of the two components) the unit ball has four extreme points.

However, in the usual category of Banach spaces with continuous linear maps as morphisms, $c\times c$ and $c$ are in fact isomorphic (because $c\cong \mathbb R \times c_0 \cong c_0\cong c_0\times c_0 \cong c\times c$, the first iso is $(x_n)\mapsto (x_\infty,x_n-x_\infty)$ where $x_\infty$ is the limit, and $c_0\times c_0\cong c_0$ by shuffeling two null sequences into a single sequence).

There might be simpler examples in this category, but I think the celebrated example of Argyros and Haydon of a Banach space $X$ where every continuous linear operator is of the form $\lambda id +K$ for a compact operator $K$ will be an example, because on $X\times X$ there are certainly other operators (the projections followed by the embeddings).