Image of Convergent Sequence in Topological Vector Space under Bounded Linear Transformation is von Neumann-Bounded
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Theorem
Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be topological vector spaces over $\GF$.
Let $T : X \to Y$ be a bounded linear transformation.
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence in $X$.
Then $\set {T x_n : n \in \N}$ is a von Neumann-bounded subset of $Y$.
Proof
Let:
- $E = \set {x_n : n \in \N}$
From Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded, $E$ is von Neumann-bounded.
So, from the definition of a bounded linear transformation:
- $T \sqbrk E = \set {T x_n : n \in \N}$ is von Neumann-bounded.
$\blacksquare$
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.32$: Theorem