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...h space in its own right. The construction is motivated by the classical [[Lp space|L^''p'' space]]s.<ref name="helemskii" /> Let <math>(X_i)_{i \in I}</math> be a family of Banach spaces, where <math>I</math> may have arbitrarily large cardinality. Set ...

...K-space''' is an [[F-space]] <math>V</math> such that every extension of F-spaces (or twisted sum) of the form The [[Lp space|<math>\ell^p</math> spaces]] for <math>0< p < 1</math> are K-spaces,<ref name="kalton"/> as are all finite dimensional [[Banach space]]s. ...

...graph theorem]] is valid for [[linear map]]s defined on and valued in most spaces encountered in analysis.{{sfn|Trèves|2006|p=549}} ...an space]] as well as many other spaces that occur in analysis are Souslin spaces. The Borel graph theorem states:{{sfn|Trèves|2006|p=549}} ...

...re [[Complete metric space|complete]] under their respective [[Lp space#Lp spaces and Lebesgue integrals|<math>p</math>-norms]]. ...space is also called a [[Cauchy space]], because sequences in such metric spaces converge if and only if they are [[Cauchy sequence|Cauchy]]. ...

...ical analysis]], most spaces which arise in practice turn out to be Banach spaces as well. == Classical Banach spaces == ...

{{About|the vector spaces of sequences and functions|the finite-dimensional vector space distance|Che ...the latter. As a Banach space they are the continuous dual of the Banach spaces <math>\ell_1</math> of absolutely summable sequences, and <math>L^1 = L^1(X ...

...set|closed]] [[linear subspace]] of the [[space of bounded sequences]], [[Lp space|<math>\ell^\infty</math>]], and contains as a closed subspace the Ban {{Banach spaces}} ...

...ef> are generalisations of the more familiar [[Lp space|<math>L^{p}</math> spaces]]. ...z spaces are denoted by <math>L^{p,q}</math>. Like the <math>L^{p}</math> spaces, they are characterized by a [[norm (mathematics)|norm]] (technically a [[q ...

...infty</math>) is the usual space of real valued functions used to define [[Lp space]]s <math>L^p,</math> then <math>\mathcal{L}^p(\mu)</math> is countabl * {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} --> ...

...ded linear operators and rearrangement invariant sequence spaces, to the [[Lp space|weak-''l''₁ sequence space]]. * [[Lp space]] ...

...as of [[mathematics]], '''distinguished spaces''' are [[topological vector spaces]] (TVSs) having the property that [[weak-* topology|weak-*]] bounded subset All [[normed space]]s and [[semi-reflexive space]]s are distinguished spaces.{{sfn|Khaleelulla|1982|pp=28-63}} ...

| contribution = Chapter V: Metric Spaces * {{annotated link|Lp space|<math>L^p</math> space}} ...

...erexample to general statements concerning the structure of general Banach spaces. The space was first introduced in 1950 in a short paper by [[Robert C. Jam ...very closed infinite-dimensional subspace contains an isomorphic copy of [[Lp space|ℓ₂]]. ...

* '''[[Lp space#Lp spaces and Lebesgue integrals|L^p convergence]]''' ...

The [[Lp space|L^p spaces]] (<math>1 \leq p \leq \infty</math>) are [[Banach lattice]]s under their c These spaces are order complete for <math>p < \infty</math>. ...

to be [[dense (topology)|dense]] in a [[Lp-space#Weighted Lp spaces|weighted L₂ space]] on the real line. It was discovered by [[Mar ...

Suppose <math>f</math> is in the [[Lp space|Lebesgue space]] <math>L^p(\Reals^d)</math> and <math>g</math> is in Here the star denotes [[convolution]], <math>L^p</math> is [[Lp space|Lebesgue space]], and ...

...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 ...

...em by [[Frigyes Riesz]] about convergence in [[Lp spaces|L^''p'' spaces]] published in 1928.<ref>{{cite journal|journal=Periodica Mathematica Hunga ...

...ainty principle]]s in the [[Fourier analysis]] of [[Lp space|L^p spaces]]. The '''(''q'',&nbsp;''p'')-norm''' of the ''n''-dimensional [[Fourier t {{Lp spaces}} ...