--- title: "FunctionSpace" language: "en" type: "Entity" summary: "Notable function spaces from functional analysis." canonical_url: "https://reference.wolfram.com/language/ref/entity/FunctionSpace.html" source: "Wolfram Language Documentation" related_guides: - title: "Entity Types" link: "https://reference.wolfram.com/language/guide/EntityTypes.en.md" --- # FunctionSpace Notable function spaces from functional analysis. Entity["FunctionSpace", name] or Entity[...] represents an entity of type "FunctionSpace". Entity[...][prop] gives the value of a specified property. Entity[...][{propi, …}] gives the value of a list of properties. EntityClass["FunctionSpace", {propi -> speci, …}] represents a class of entities with values of propi defined by speci. ## Sample Entities * Entity["FunctionSpace", {{"LebesgueL", {{"Reals", \[FormalN]}, {"LebesgueMeasure", \[FormalN]}}}, "Infinity"}]Entity["FunctionSpace", {{"ContinuouslyDifferentiableC", {"Reals", \[FormalN]}}, "Infinity"}]Entity["FunctionSpace", {{"ContinuouslyDifferentiableC", {"Reals", \[FormalN]}}, 0}]Entity["FunctionSpace", {{"ContinuouslyDifferentiableC", {"ClosedInterval", {\[FormalA], \[FormalB]}}}, \[FormalK]}]Entity["FunctionSpace", {{"ContinuouslyDifferentiableC", {"ClosedInterval", {0, 1}}}, 2}]Entity["FunctionSpace", {{"GeneralBesovB", {{"Reals", \[FormalN]}, {"LebesgueMeasure", \[FormalN]}}}, {\[FormalP], \[FormalQ], \[FormalS]}}]Entity["FunctionSpace", {{"AnalyticQ", {"ComplexOpenUnitDisk", {"NormalizedLebesgueMeasure", 2}}}, \[FormalP]}]Entity["FunctionSpace", {{"HoelderC", {"Reals", \[FormalN]}}, {\[FormalK], \[FormalA]}}]Entity["FunctionSpace", {{"HolomorphicHardyH", "ComplexOpenUnitDisk"}, "Infinity"}]Entity["FunctionSpace", {{"LorentzL", {{"Reals", \[FormalN]}, {"LebesgueMeasure", \[FormalN]}}}, {\[FormalP], "Infinity"}}]**…** ## Sample Entity Classes * EntityClass["FunctionSpace", "Approximation"]EntityClass["FunctionSpace", "Continuous"]EntityClass["FunctionSpace", "Distribution"]EntityClass["FunctionSpace", "GeneralizedBergman"]EntityClass["FunctionSpace", "GeneralizedBloch"]EntityClass["FunctionSpace", "Harmonic"]EntityClass["FunctionSpace", "Holomorphic"]EntityClass["FunctionSpace", "Measurable"]EntityClass["FunctionSpace", "MorreyCampanato"]EntityClass["FunctionSpace", "TemperedDistribution"]**…** ## Properties | | | | -------------------------- | ---------------------------- | | AdditionalPeople | additional people involved | | AlternateNames | alternate names | | AlternateNotations | alternate notations | | AssociatedPeople | associated people | | BesselInequality | Bessel inequality | | CauchySchwarzInequality | Cauchy‐Schwarz inequality | | Classes | classes | | Classifications | classifications | | Dimension | dimension | | DualSpace | dual space | | DualSpaceMap | dual space map | | EmbeddingSpace | embedding space | | EntityClasses | entity classes | | EntityTypeList | entity type list | | EponymousPeople | eponymous people | | EquivalenceRelation | equivalence relation | | FormattedASCIIName | formatted ASCII name | | FormattedReferences | references | | FormulationDate | formulation date | | Formulators | formulators | | HoelderTypeInequality | Hölder‐type inequality | | InnerProduct | inner product | | IsomorphicSpaces | isomorphic spaces | | MeasureSpace | measure space | | Name | name | | NonorthonormalBasisExample | nonorthonormal basis example | | Norm | norm | | OrthonormalBasisExample | orthonormal basis example | | ParameterRestrictions | parameter restrictions | | Parameters | parameters | | PlaintextName | plaintext name | | PreDualSpace | pre‐dual space | | PreDualSpaceMap | pre‐dual space map | | QuasiNorm | quasi‐norm | | References | reference citations | | RelatedResults | related results | | RelatedSpaces | related spaces | | RelationshipGraph | relationship graph | | SampleMembers | sample members | | SampleNonmembers | sample nonmembers | | Seminorms | seminorms | | ShortNotation | short notation | | Timeline | timeline | | TriangleInequality | triangle inequality | | TypesetDescription | typeset description | | TypesetNotationsTable | typeset notations table | | UnderlyingSpace | underlying space | ## Details * ``"FunctionSpace"`` entities include notable named function spaces parametrized based on domain and measure. * ``"FunctionSpace"`` entity classes include classifications based on domain and mathematical properties. * Some properties are available for the ``"FunctionSpace"`` entity type as a whole and can be given using the form ``EntityValue["FunctionSpace", property]``. Such properties include: | | | | ----------------------------- | ------------------------------------------------------------------ | | "Properties" | the list of available properties | | "PropertyCanonicalNames" | the standard names of available properties | | "SampleEntities" | a sample list of available entities (typically of length 10) | | "SampleEntityClasses" | a sample list of available entity classes (typically of length 10) | | "EntityCount" | number of entities available | | "Entities" | the list of available entities | | "EntityCanonicalNames" | the standard names of available entities | | "EntityClasses" | the list of available entity classes | | "EntityClassCanonicalNames" | the standard names of available entity classes | | "PropertyClasses" | the list of available property classes | | "PropertyClassCanonicalNames" | the standard names of available property classes | | "PropertyCount" | number of properties available | * The following annotations can be used in the third argument of ``EntityValue["FunctionSpace", property, annotation]`` : | | | | --- | --- | | "Source" | source information for the property | | "Date" | the date associated with the entity-property value (if any) | | "EntityAssociation" | an association of entities and entity-property values | | "PropertyAssociation" | an association of properties and entity-property values | | "EntityPropertyAssociation" | an association in which the specified entities are keys, and values are a nested association of properties and entity-property values | | "PropertyEntityAssociation" | an association in which the specified properties are keys, and values are a nested association of entities and entity-property values | | "Dataset" | a dataset in which the specified entities are keys, and values are an association of property names and entity-property values | * The following annotations can be used in the second argument of ``EntityValue[property, annotation]`` : | | | | ------------------------ | --------------------------------------------------------------- | | "Qualifiers" | the list of possible qualifiers for the property | | "QualifierValues" | the list of possible values that can be given to each qualifier | | "DefaultQualifierValues" | the list of default values for the property's qualifiers | | "Description" | a brief textual description of the property | | "Definition" | a detailed textual definition of the property | | "PhysicalQuantity" | the physical quantity associated with the entity-property value | | "Unit" | the unit associated with the entity-property value | * A qualifier value of ``Automatic`` indicates that an applicable format of values can be used; e.g. for the ``"Date"`` qualifier, this includes a proper date or date span. --- ## Examples (4) ### Basic Examples (4) Use ctrl+= for entity discovery: ```wl In[1]:= \[FreeformPrompt]["lebesgue space L^2(R^n)"] Out[1]= Entity["FunctionSpace", {{"LebesgueL", {{"Reals", n}, {"LebesgueMeasure", n}}}, 2}] ``` --- Find a property value for an entity: ```wl In[1]:= Entity["FunctionSpace", {{"LebesgueL", {{"Reals", n}, {"LebesgueMeasure", n}}}, 2}]["Norm"] Out[2]= Function[\[FormalF], √Inactive[Integrate][Abs[\[FormalF][\[FormalX]]]^2, {\[FormalX], PureMath`MeasureSpace[ℝ^n, PureMath`LebesgueMeasure[n]]}]] ``` --- Retrieve a dataset of all available properties for an entity: ```wl In[1]:= Entity["FunctionSpace", {{"LorentzL", {{"Reals", \[FormalN]}, {"LebesgueMeasure", \[FormalN]}}}, {\[FormalP], "Infinity"}}]["Dataset"] Out[1]= Dataset[Association[EntityProperty["FunctionSpace", "AdditionalPeople"] -> Missing["NotAvailable"], EntityProperty["FunctionSpace", "AlternateNames"] -> Missing["NotAvailable"], EntityProperty["FunctionSpace", "AlternateNotations"] -> {Su ... Row[{"Lorentz space ", Superscript[Style["L", Italic], Row[{Style["\[FormalP]", Italic], ",∞"}]], Row[{RawBoxes["("], Superscript["\[DoubleStruckCapitalR]", Style["\[FormalN]", Italic]], ", ", "\[DifferentialD]", Superscript[Style["x", Italic], Style["\[FormalN]", Italic]], RawBoxes[")"]}]}]]] ``` --- Retrieve tabulated function spaces that are measurable Hilbert spaces: ```wl In[1]:= EntityClass["FunctionSpace", {"Classifications" -> ContainsAll[{"HilbertSpace"}], "Classes" -> ContainsAll[{"Measurable"}]}]//EntityList Out[1]= {Entity["FunctionSpace", {"BargmannSegalFockF", {{"Complexes", \[FormalN]}, {"ComplexLebesgueMeasure", \[FormalN]}}}], Entity["FunctionSpace", {{"BergmanA", {"ComplexOpenUnitDisk", {"NormalizedLebesgueMeasure", 2}}}, 2}], Entity["FunctionSpace", {{"HarmonicBergmanA" ... sitiveIntegers", "CountingMeasure"}}, 2}], Entity["FunctionSpace", {{"SobolevW", {{"Reals", \[FormalN]}, {"LebesgueMeasure", \[FormalN]}}}, {2, \[FormalK]}}], Entity["FunctionSpace", {{"WeightedBergmanA", {"ComplexOpenUnitDisk", {"NormalizedLebesgueMeasure", 2}}}, {2, \[FormalAlpha]}}]} ``` ## See Also * [`MathematicalFunction`](https://reference.wolfram.com/language/ref/entity/MathematicalFunction.en.md) * [`FunctionalAnalysisSource`](https://reference.wolfram.com/language/ref/entity/FunctionalAnalysisSource.en.md) * [`FunctionSpace`](https://reference.wolfram.com/language/ref/textcontent/FunctionSpace.en.md) ## Related Guides * [Entity Types](https://reference.wolfram.com/language/guide/EntityTypes.en.md) ## History * [Introduced in 2019 (12.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn120.en.md)