Image of Convergent Sequence in Topological Vector Space under Bounded Linear Transformation is von Neumann-Bounded

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