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...''', named after [[Issai Schur]], is the property of [[normed vector space|normed space]]s that is satisfied precisely if [[weak topology|weak convergence]] When we are working in a normed space ''X'' and we have a sequence <math>(x_{n})</math> that converges weak ...

Let <math>(X_i)_{i \in I}</math> be a family of Banach spaces, where <math>I</math> may have arbitrarily large cardinality. Set The result is a [[normed space|normed]] Banach space, and this is precisely the L^''p'' sum of <math>(X ...

...roperty of [[Euclidean space]]s holds for general [[finite-dimensional]] [[normed vector space]]s. Let (''V'', ||·||) be an ''n''-dimensional normed vector space. Then there exists a [[basis (linear algebra)|basis]] {''e''<s ...

Let <math>X</math> be a [[Normed space|normed]] vector space. If <math>C \subseteq X</math> is a [[Convex set|convex]] [[ * {{Zălinescu Convex Analysis in General Vector Spaces 2002}} <!-- {{sfn|Zălinescu|2002|pp=}} --> ...

The '''Radon–Riesz property''' is a mathematical property for [[normed space]]s that helps ensure [[limit of a sequence|convergence]] in norm. Giv Suppose that (''X'', ||·||) is a normed space. We say that ''X'' has the ''Radon–Riesz property'' (or that ''X'' is ...

...ance, the set of [[isometry]] classes of <math>n</math>-dimensional normed spaces becomes a [[compact metric space]], called the '''Banach–Mazur compactum''' If <math>X</math> and <math>Y</math> are two finite-dimensional normed spaces with the same dimension, let <math>\operatorname{GL}(X, Y)</math> denote th ...

...very isometry of a [[normed vector space|normed real linear space]] onto a normed real linear space is a [[linear mapping]] up to translation. In 1970, [[Ale ...sandrov–Rassias Problem.''' If {{mvar|X}} and {{mvar|Y}} are normed linear spaces and if {{math|''T'' : ''X'' → ''Y''}} is a continuous and/or surjective map ...

...r, there exist [[Semi-reflexive space|semi-reflexive]] countably barrelled spaces that are not barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}} There exist σ-barrelled spaces that are not countably barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}} ...

...ical space]]s, [[uniform space]]s, [[topological abelian group]]s (TAG), [[normed vector space]]s, [[Euclidean space]]s, and the [[Real number|real]]/[[Compl ===...in a normed space (''N'')=== ...

...a property similar to the definition of [[totally bounded]] subsets. These spaces were introduced by [[Alexander Grothendieck]]. * Vector subspace of Schwartz spaces are Schwartz spaces. ...

In mathematics, '''amalgam spaces''' categorize functions with regard to their local and global behavior. Wh ...\|\cdot \|_X </math>. Then the ''Wiener amalgam space''<ref>Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis by Hans Georg Feichtinger an ...

...om {{mvar|X}} to {{mvar|Y}}, where if {{mvar|X}} and {{mvar|Y}} are normed spaces then we endow {{math|L(''X''; ''Y'')}} with its canonical [[operator norm]] ...inear extension]] to all of {{mvar|X}}. If {{mvar|X}} and {{mvar|Y}} are [[normed space]]s, then we say that {{mvar|Y}} has '''the metric extension property ...

In [[mathematics]], a '''uniformly smooth space''' is a [[normed vector space]] <math>X</math> satisfying the property that for every <math> The '''modulus of smoothness''' of a normed space ''X'' is the function &rho;_''X'' defined for every {{nowra ...

..., a '''Banach lattice''' {{math|(''X'',‖·‖)}} is a [[Banach space|complete normed vector space]] with a [[lattice order]], <math>\leq</math>, such that for a Examples of non-lattice Banach spaces are now known; [[James space|James' space]] is one such.<ref>Kania, Tomasz ...

In mathematics, '''cocompact embeddings''' are [[embedding]]s of [[normed vector space]]s possessing a certain property similar to but weaker than [[ Let <math>G</math> be a group of isometries on a normed vector space <math>X</math>. One says that a sequence <math>(x_k)\subset X ...

...rt description|Generalization of inner products that applies to all normed spaces}} ...for the purpose of extending [[Hilbert space]] type arguments to [[Banach spaces]] in [[functional analysis]].<ref name=":0">{{citation ...

{{Short description|All infinite-dimensional, separable Banach spaces are homeomorphic}} ...r, more generally, [[Fréchet space]]s, are [[homeomorphic]] as topological spaces. The theorem was proved by [[Mikhail Kadets|Mikhail Kadec]] (1966) and [[Ri ...

==Lebesgue spaces== ...articularly studied for [[Lp space|Lebesgue spaces]], finite-dimensional [[normed vector space]]s with the <math>L^p</math> norm ...

is finite. The set of such sequences forms a [[normed space]] with the [[vector space]] operations defined [[Componentwise operat * {{annotated link|List of Banach spaces}} ...

...nalogous to the class of locally convex topological vector spaces over the normed fields of real or complex numbers in functional analysis."<ref name="exact" ...1=Positselski |first1=Leonid |title=Exact categories of topological vector spaces with linear topology |journal=Moscow Math. Journal |date=2024 |volume=24 |i ...