Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded/Proof 2
Jump to navigation
Jump to search
Theorem
Let $\struct {X, \tau}$ be a topological vector space.
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence with $x_n \to x$.
Let:
- $E = \set {x_n : n \in \N}$
Then $E$ is von Neumann-bounded.
Proof
From Union of Image of Convergent Sequence and Limit in Topological Space is Compact, $E \cup \set x$ is compact.
From Compact Subspace of Topological Vector Space is von Neumann-Bounded, $E \cup \set x$ is von Neumann bounded.
From Subset of von Neumann-Bounded Set is von Neumann-Bounded, $E$ is von Neumann bounded.
$\blacksquare$