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A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.
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Grothendieck on topological vector spaces
Maybe you will find interesting the following references
Grothendieck's Theorem, past and present, Pisier
or
Grothendieck’s works on Banach spaces and
their surprising recent repercussions
A. Grothend …