I want to prove that a product $\prod_{i\in I}X_i$ of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial topology. Could anyone point me in the right direction?
Pseudonormable Product Spaces
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general-topology
functional-analysis
metric-spaces
normed-spaces
topological-vector-spaces
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0Alright. Thanks! – 2012-12-27
1 Answers
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I guess this is a simple combination of the following well-known facts:
A TVS $X$ is seminormable (or, as called above, pseudonormable) iff it is locally convex and has a bounded $0$-neighbourhood (i.e. absorbed by every other $0$-neighbourhood).
A set $B$ is bounded in a product space $\prod_i X_i$ with product topology iff $B=\prod_i B_i$ where each $B_i$ is bounded in $X_i$
If a TVS $X$ is bounded, its topology is the trivial topology (i.e. $\emptyset$ and $X$ are the only open sets).
So if $\prod_i X_i$ is seminormable, then there is a $0$-neighbourhood $B=\prod_i B_i$ where each $B_i$ is bounded in $X_i$ and hence for all but at most finitely many $i$ you have $B_i = X_i$.