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- mathematischer Satz (de)
- théorème d'analyse fonctionnelle (fr)
- la teoremo, ke ĉiupunkte barita aro da linearaj operatoroj inter banaĥaj spacoj estas unuforme barita laŭ la operatora normo (eo)
- enunciado que afirma que un conjunto de operadores lineales acotados puntualmente en un espacio de Banach está acotado uniformemente en la norma del operador (es)
- משפט מתמטי יסודי באנליזה פונקציונלית (iw)
- theorem that a pointwise bounded set of linear operators on a Banach space is uniformly bounded in operator norm (en)
- izrek, da je točkovno omejena množica linearnih operatorjev na Banachovem prostoru enakomerno omejena v normi operatorja (sl)
- twierdzenie analizy funkcjonalnej o ciągach operatorów liniowych (pl)
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- Let be a set of continuous linear operators between two topological vector spaces and .
For every denote the orbit of by
and let denote the set of all whose orbit is a bounded subset of
If is of the second category in then and is equicontinuous. (en)
- If a sequence of bounded operators converges pointwise, that is, the limit of exists for all then these pointwise limits define a bounded linear operator (en)
- If is a sequence of continuous linear maps from an F-space into a Hausdorff topological vector space such that for every the limit
exists in then is a continuous linear map and the maps are equicontinuous. (en)
- Suppose that is a sequence of continuous linear maps between two topological vector spaces and
# If the set of all for which is a Cauchy sequence in is of the second category in then
# If the set of all at which the limit exists in is of the second category in and if is a complete metrizable topological vector space , then and is a continuous linear map. (en)
- Any weakly bounded subset in a normed space is bounded. (en)
- Let be a set of continuous linear operators between two topological vector spaces and and let be any bounded subset of
Then the family of sets is uniformly bounded in if any of the following conditions are satisfied:
# is equicontinuous.
# is a convex compact Hausdorff subspace of and for every the orbit is a bounded subset of (en)
- Let be a set of continuous linear operators from a complete metrizable topological vector space into a Hausdorff topological vector space
If for every the orbit
is a bounded subset of then is equicontinuous.
So in particular, if is also a normed space and if
then is equicontinuous. (en)
- Let be a Banach space, a normed vector space and the space of all continuous linear operators from into . Suppose that is a collection of continuous linear operators from to If, for every ,
then (en)
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- Proposition (en)
- Corollary (en)
- Theorem (en)
- Uniform Boundedness Principle (en)
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- Let be balanced neighborhoods of the origin in satisfying It must be shown that there exists a neighborhood of the origin in such that for every
Let
which is a closed subset of that for every also satisfies and
.
If then being bounded in implies that there exists some integer such that so if then
Since was arbitrary,
This proves that
Because is of the second category in the same must be true of at least one of the sets for some
The map defined by is a homeomorphism, so the set is necessarily of the second category in
Because is closed and of the second category in its topological interior in is not empty.
Pick
Because the map defined by is a homeomorphism, the set
is a neighborhood of in which implies that the same is true of its superset
And so for every
This proves that is equicontinuous.
Q.E.D.
Because is equicontinuous, if is bounded in then is uniformly bounded in
In particular, for any because is a bounded subset of is a uniformly bounded subset of Thus
Q.E.D. (en)
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- Uniform boundedness principle (en)
- Teorema de Banach-Steinhaus (ca)
- Banachova–Steinhausova věta (cs)
- Satz von Banach-Steinhaus (de)
- Principio dell'uniforme limitatezza (it)
- Théorème de Banach-Steinhaus (fr)
- 균등 유계성 원리 (ko)
- Twierdzenie Banacha-Steinhausa (pl)
- Principe van uniforme begrensdheid (nl)
- Teorema de Banach-Steinhaus (pt)
- Принцип равномерной ограниченности (ru)
- Banach-Steinhaus sats (sv)
- Теорема Банаха — Штейнгауза (uk)
- 一致有界性原理 (zh)
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