Search results

#REDIRECT [[Lp space#Lp spaces and Lebesgue integrals]] ...

#REDIRECT [[Lp_space#Lp spaces and Lebesgue integrals]] ...

* The classical function spaces [[Lp space|L^''p'' and <math>\ell^p</math>]] ...

...due to [[Kurt O. Friedrichs|Kurt Friedrichs]]. It places a bound on the [[Lp space|''L^p'' norm]] of a function using ''L^p'' bounds * <math>\| \cdot \|_{L^p (\Omega)}</math> denotes the [[Lp space|''L^p'' norm]]; ...

==Important classes of measure spaces== Most important classes of measure spaces are defined by the properties of their associated measures. This includes, ...

==Common measurable spaces== ...e space <math>(X, \mathcal B(X))</math> that is common for all topological spaces such as the real numbers <math>\R.</math> ...

For the <math>\ell^p</math> [[lp space|spaces]], the ''P''_''n'' is reflexive if and only if {{mvar|n}} < {{mva ...ace]], it is not polynomially reflexive. This makes polynomially reflexive spaces rare. ...

...up> and ''L''^''q'' are [[dual space|dual]] to each other (see [[Lp space|''L''^''p'' space]]). ...

...ous function|continuous]] function between [[topological space|topological spaces]] [[Morphism|preserves]] the topological structure: the preimage of any [[o Let <math>(X,\Sigma)</math> and <math>(Y,\Tau)</math> be measurable spaces, meaning that <math>X</math> and <math>Y</Math> are sets equipped with resp ...

{{for|F-spaces in general topology|sub-Stonean space}} ...math>X</math> is [[Continuous function#Continuous functions between metric spaces|continuous]] with respect to <math>d</math> and the standard metric on <mat ...

...able]], but only [[Integrable function|integrable]], i.e., to lie in the [[Lp space|L^''p'' space]] <math>L^1([a,b])</math>. Let <math>u</math> be a function in the [[Lp space|Lebesgue space]] <math>L^1([a,b])</math>. We say that <math>v</math> ...

{{Short description|Type of convergence in Hilbert spaces}} ...that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an [[orthonormal basis]] in an infinite-dim ...

...ces]], which resembles the [[Lebesgue measure]] used in finite-dimensional spaces. ...-like measures can exist in specific contexts. These include non-separable spaces like the [[Hilbert cube]], or scenarios where some typical properties of fi ...

{{See also|Vector space#Function spaces}} ...structure. For example, if {{var|V}} and also {{var|X}} itself are vector spaces over {{var|F}}, the set of [[Linear map|linear maps]] {{var|X}} → {{var|V}} ...

{{Short description|Class of Banach spaces}} All three spaces are complete (they are [[Banach space]]s) with respect to the same norm def ...

* [[Fundamental theorem of Hilbert spaces]] *[[Lp space]] ...

...math>f</math> is a [[square-integrable function]] or, more generally, in [[Lp space|''L''^''p'' space]] <math>L^2[-\pi, \pi].</math> A similar ...generally, Parseval's identity holds for arbitrary [[Hilbert space|Hilbert spaces]], not necessarily separable. When the Hilbert space is not separable any o ...

...tor ''A'' is defined everywhere (and, in turn, the completeness of Hilbert spaces). ...instance the [[quantum harmonic oscillator]]. Here the Hilbert space is [[Lp space|L]]²('''R'''), the space of square integrable functions on ...

Given a [[measure space]] <math>(S, \Sigma, \mu),</math> the [[Lp space|space <math>\mathcal{L}^\infty(S, \mu)</math>]] consisting of all of * {{annotated link|Lp space|<math>L^p</math> space}} ...

...al |last=Karno |first=Zbigniew |date=1994 |title=On Kolmogorov Topological Spaces |url=https://mizar.uwb.edu.pl/JFM/pdf/tsp_1.pdf |journal=Journal of Formali ...usdorff space|T₂ (or Hausdorff) spaces]], i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction ...