Category:Banach-Steinhaus Theorem
This category contains pages concerning Banach-Steinhaus Theorem:
Normed Vector Space
Let $\struct {X, \norm {\,\cdot\,}_X}$ be a Banach space.
Let $\struct {Y, \norm {\,\cdot\,}_Y}$ be a normed vector space.
Let $\family {T_\alpha: X \to Y}_{\alpha \mathop \in A}$ be an $A$-indexed family of bounded linear transformations from $X$ to $Y$.
Suppose that:
- $\ds \forall x \in X: \sup_{\alpha \mathop \in A} \norm {T_\alpha x}_Y$ is finite.
Then:
- $\ds \sup_{\alpha \mathop \in A} \norm {T_\alpha}$ is finite
where $\norm {T_\alpha}$ denotes the norm of the linear transformation $T_\alpha$.
Topological Vector Space
Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be topological vector spaces over $\GF$.
Let $\Gamma$ be a set of continuous linear transformations $X \to Y$.
Let $B$ be the set of all $x \in X$ such that:
- $\map \Gamma x = \set {T x : T \in \Gamma}$
is von Neumann-bounded in $Y$.
Suppose that $B$ is not meager in $X$.
Then $B = X$ and $\Gamma$ is equicontinuous.
$F$-Space
Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau_X}$ be an $F$-Space over $\GF$.
Let $\struct {Y, \tau_Y}$ be a topological vector space over $\GF$.
Let $\Gamma$ be a set of continuous linear transformations $X \to Y$ such that for all $x \in X$:
- $\map \Gamma x = \set {T x : T \in \Gamma}$ is von Neumann-bounded in $Y$.
Then $\Gamma$ is equicontinuous.
Pages in category "Banach-Steinhaus Theorem"
The following 10 pages are in this category, out of 10 total.
B
- Banach-Steinhaus Theorem
- Banach-Steinhaus Theorem/F-Space
- Banach-Steinhaus Theorem/Normed Vector Space
- Banach-Steinhaus Theorem/Normed Vector Space/Also known as
- Banach-Steinhaus Theorem/Normed Vector Space/Proof 1
- Banach-Steinhaus Theorem/Normed Vector Space/Proof 2
- Banach-Steinhaus Theorem/Normed Vector Space/Proof 3
- Banach-Steinhaus Theorem/Topological Vector Space