Definition:Bounded Linear Transformation/Topological Vector Space

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Let $\GF \in \set {\R, \C}$.

Let $X$ and $Y$ be topological vector spaces over $\GF$.

Let $T : X \to Y$ be a linear transformation.

We say that $T$ is a bounded linear transformation if and only if:

for each von Neumann-bounded subset $E$ of $X$, $T \sqbrk E$ is von Neumann-bounded.