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((Navbox
| name = Banach spaces
| state = (({state|(({1|))))))
| title = [[Banach space]] topics
| listclass = hlist

| group1 = Types of Banach spaces
| list1 =
* [[Asplund space|Asplund]]
* [[Banach space|Banach]]
** [[List of Banach spaces|list]]
* [[Banach lattice]]
* [[Grothendieck space|Grothendieck ]]
* [[Hilbert space|Hilbert]]
** [[Inner product space]]
** [[Polarization identity]]
* ([[Polynomially reflexive space|Polynomially]])&nbsp;[[Reflexive space|Reflexive]]
* [[Riesz space|Riesz]]
* [[L-semi-inner product]]
* ([[B-convex space|B]]
*   [[Strictly convex space|Strictly]]
*   [[Uniformly convex space|Uniformly]])&nbsp;convex
* [[Uniformly smooth space|Uniformly smooth]]
* ([[Injective tensor product|Injective]]
*   [[Projective tensor product|Projective]])&nbsp;[[Topological tensor product|Tensor product]]&nbsp;([[Tensor product of Hilbert spaces|of Hilbert spaces]])

| group2 = Banach spaces are:
| list2 = 
* [[Barrelled space|Barrelled]]
* [[Complete topological vector space|Complete]]
* [[F-space]]
* [[Fréchet space|Fréchet]]
** [[Differentiation in Fréchet spaces#Tame Fréchet spaces|tame]]
* [[Locally convex topological vector space|Locally convex]]
** [[Locally convex topological vector space#Definition via seminorms|Seminorm]]s/[[Minkowski functional]]s
* [[Mackey space|Mackey]]
* [[Metrizable topological vector space|Metrizable]]
* [[Normed space|Normed]]
** [[Norm (mathematics)|norm]]
* [[Quasinorm|Quasinormed]]
* [[Stereotype space|Stereotype]]

| group3 = Function space Topologies
| list3 =
* [[Banach–Mazur compactum]]
* [[Dual topology|Dual]]
* [[Dual space]]
** [[Dual norm]]
* [[Operator topologies|Operator]]
* [[Ultraweak topology|Ultraweak]]
* [[Weak topology|Weak]]
** [[Weak topology (polar topology)|polar]]
** [[Weak operator topology|operator]]
* [[Strong topology|Strong]]
** [[Strong topology (polar topology)|polar]]
** [[Strong operator topology|operator]]
* [[Ultrastrong topology|Ultrastrong]]
* [[Topology of uniform convergence|Uniform convergence]]

| group4 = [[Linear operator]]s
| list4 =
* [[Hermitian adjoint|Adjoint]]
* [[Bilinear map|Bilinear]]
** [[Bilinear form|form]]
** [[Bilinear map|operator]]
** [[Sesquilinear form|sesquilinear]]
* ([[Unbounded operator|Un]])[[Bounded operator|Bounded]]
* [[Closed linear operator|Closed]]
* [[Compact operator|Compact]]
** [[Compact operator on Hilbert space|on Hilbert spaces]]
* ([[Discontinuous linear map|Dis]])[[Continuous linear operator|Continuous]]
* [[Densely defined]]
* Fredholm
** [[Fredholm kernel|kernel]]
** [[Fredholm operator|operator]]
* [[Hilbert–Schmidt operator|Hilbert–Schmidt]]
* [[Linear form|Functionals]]
** [[Positive linear functional|positive]]
* [[Pseudo-monotone operator|Pseudo-monotone]]
* [[Normal operator|Normal]]
* [[Nuclear operator|Nuclear]]
* [[Self-adjoint operator|Self-adjoint]]
* [[Strictly singular operator|Strictly singular]]
* [[Trace class]]
* [[Transpose of a linear map|Transpose]]
* [[Unitary operator|Unitary]]

| group5 = [[Operator theory]]
| list5 =
* [[Banach algebra]]s
* [[C*-algebra]]s
* [[Operator space]]
* [[Spectrum (functional analysis)|Spectrum]]
** [[Spectrum of a C*-algebra|C*-algebra]]
** [[Spectral radius|radius]]
* [[Spectral theory]]
** [[Spectral theory of ordinary differential equations|of ODEs]]
** [[Spectral theorem]]
* [[Polar decomposition]]
* [[Singular value decomposition]]

|group6 = ((Cl|Theorems in functional analysis|Theorems))
|list6 =
* [[Anderson–Kadec theorem|Anderson–Kadec]]
* [[Banach–Alaoglu theorem|Banach–Alaoglu]]
* [[Banach–Mazur theorem|Banach–Mazur]]
* [[Banach–Saks theorem|Banach–Saks]]
* [[Open mapping theorem (functional analysis)|Banach–Schauder (open mapping)]]
* [[Uniform boundedness principle|Banach–Steinhaus (Uniform boundedness)]]
* [[Bessel's inequality]]
* [[Cauchy–Schwarz inequality]]
* [[Closed graph theorem|Closed graph]]
* [[Closed range theorem|Closed range]]
* [[Eberlein–Šmulian theorem|Eberlein–Šmulian]]
* [[Freudenthal spectral theorem|Freudenthal spectral]]
* [[Gelfand–Mazur theorem|Gelfand–Mazur]]
* [[Gelfand–Naimark theorem|Gelfand–Naimark]]
* [[Goldstine theorem|Goldstine]]
* [[Hahn–Banach theorem|Hahn–Banach]]
** [[Hyperplane separation theorem|hyperplane separation]]
* [[Kakutani fixed-point theorem#Infinite-dimensional generalizations|Kakutani fixed-point]]
* [[Krein–Milman theorem|Krein–Milman]]
* [[Invariant subspace problem#Known special cases|Lomonosov's invariant subspace]]
* [[Mackey–Arens theorem|Mackey–Arens]]
* [[Mazur's lemma]]
* [[M. Riesz extension theorem|M. Riesz extension]]
* [[Parseval's identity]]
* [[Riesz's lemma]]
* [[Riesz representation theorem|Riesz representation]]
* [[Ursescu theorem#Robinson–Ursescu theorem|Robinson-Ursescu]]
* [[Schauder fixed-point theorem|Schauder fixed-point]]

| group7 = Analysis
| list7  =
* [[Abstract Wiener space]]
* [[Banach manifold]]
** [[Banach bundle|bundle]]
* [[Bochner space]]
* [[Convex series]]
* [[Differentiation in Fréchet spaces]]
* [[Derivative]]s
** [[Fréchet derivative|Fréchet]]
** [[Gateaux derivative|Gateaux]]
** [[Functional derivative|functional]]
** [[Infinite-dimensional holomorphy|holomorphic]]
** [[Quasi-derivative|quasi]]
* [[Integral]]s
** [[Bochner integral|Bochner]]
** [[Dunford integral|Dunford]]
** [[Pettis integral|Gelfand–Pettis]]
** [[Regulated integral|regulated]]
** [[Paley–Wiener integral|Paley–Wiener]]
** [[Pettis integral|weak]]
* [[Functional calculus]]
** [[Borel functional calculus|Borel]]
** [[Continuous functional calculus|continuous]]
** [[Holomorphic functional calculus|holomorphic]]
* [[Measure (mathematics)|Measures]]
** [[Infinite-dimensional Lebesgue measure|Lebesgue]]
** [[Projection-valued measure|Projection-valued]]
** [[Vector measure|Vector]]
* [[Weakly measurable function|Weakly]] / [[Strongly measurable functions|Strongly]] measurable function

| group8 = Types of sets
| list8 =
* [[Absolutely convex set|Absolutely convex]]
* [[Absorbing set|Absorbing]]
* [[Affine space|Affine]]
* [[Balanced set|Balanced/Circled]]
* [[Bounded set (topological vector space)|Bounded]]
* [[Convex set|Convex]]
* [[Convex cone|Convex cone ((smaller|(subset)))]]
* [[Convex series#Types of subsets|Convex series related]]&nbsp;((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H''x''), and (Hw''x''))
* [[Cone (linear algebra)|Linear cone ((smaller|(subset)))]]
* [[Radial set|Radial]]
* [[Star domain|Radially convex/Star-shaped]]
* [[Symmetric set|Symmetric]]
* [[Zonotope]]

| group9 = Subsets((\))set operations
| list9 =
* [[Affine hull]]
* ([[Algebraic interior#Relative algebraic interior|Relative]])&nbsp;[[Algebraic interior|Algebraic interior (core)]]
* [[Bounding point]]s
* [[Convex hull]]
* [[Extreme point]]
* [[Interior (topology)|Interior]]
* [[Linear span]]
* [[Minkowski addition|Minkowski addition]]
* [[Polar set|Polar]]
* ([[Algebraic interior#Quasi relative interior|Quasi]])&nbsp;[[Algebraic interior#Relative interior|Relative interior]]

| group10 = [[:Template:ListOfBanachSpaces|Examples]]
| list10 =
* [[Absolute continuity|Absolute continuity ''AC'']]
* [[ba space|<math>ba(\Sigma)</math>]]
* [[c space]]
* [[BK-space|Banach coordinate ''BK'']]
* [[Besov space|Besov <math>B^s_{p,q}(\R)</math>]]
* [[Birnbaum–Orlicz space|Birnbaum–Orlicz]]
* [[Bounded variation|Bounded variation ''BV'']]
* [[Bs space]]
* [[Continuous functions on a compact Hausdorff space|Continuous ''C(K)'' with ''K'' compact Hausdorff]]
* [[Hardy space|Hardy H<sup>''p''</sup>]]
* [[Hilbert space#Definition|Hilbert ''H'']]
* [[Morrey–Campanato space|Morrey–Campanato <math>L^{\lambda, p}(\Omega)</math>]]
* [[Sequence space#ℓp spaces|''ℓ<sup>p</sup>'']]
** [[L-infinity#Sequence space|<math>\ell^{\infty}</math>]]
* [[Lp space|''L<sup>p</sup>'']]
** [[L-infinity#Function space|<math>L^\infty</math>]]
** [[Lp space#Weighted Lp spaces|weighted]]
* [[Schwartz space|Schwartz <math>S \left(\R^n\right)</math>]]
* [[Segal–Bargmann space|Segal–Bargmann ''F'']]
* [[Sequence space]]
* [[Sobolev space|Sobolev W<sup>''k,p''</sup>]]
** [[Sobolev inequality]]
* [[Triebel–Lizorkin space|Triebel–Lizorkin]]
* [[Wiener amalgam space|Wiener amalgam <math>W(X,L^p)</math>]]

| group11 = Applications
| list11 =
* [[Differential operator]]
* [[Finite element method]]
* [[Mathematical formulation of quantum mechanics]]
* [[Spectral theory of ordinary differential equations|Ordinary Differential Equations (ODEs)]]
* [[Validated numerics]]

))<noinclude>
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</noinclude>

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