View source for Glossary of functional analysis

{{See also|List of functional analysis topics}}
This is a [[glossary]] for the terminology in a mathematical field of [[functional analysis]].

Throughout the article, unless stated otherwise, the base field of a vector space is the field of real numbers or that of complex numbers. Algebras are not assumed to be unital.

{{Compact ToC|short1|sym=yes|x=[[#XYZ|XYZ]]|y=|z=|seealso=yes|refs=yes}}

See also: [[List of Banach spaces]], [[glossary of real and complex analysis]].

==*==
{{glossary}}

{{term|1=*}}
{{defn|1=[[*-homomorphism]] between [[involutive Banach algebra]]s is an algebra homomorphism preserving *.}}

{{glossary end}}

==A==
{{glossary}}

{{term|1=abelian}}
{{defn|1=Synonymous with "commutative"; e.g., an abelian Banach algebra means a commutative Banach algebra.}}

{{term|1=Anderson–Kadec}}
{{defn|1=The [[Anderson–Kadec theorem]] says a separable infinite-dimensional Fréchet space is isomorphic to <math>\mathbb{R}^{\mathbb{N}}</math>.}}

{{term|1=Alaoglu}}
{{defn|1=[[Alaoglu's theorem]] states that the closed unit ball in a normed space is compact in the [[weak-* topology]].}}

{{term|1=adjoint}}
{{defn|1=The [[adjoint operator|adjoint]] of a bounded linear operator <math>T: H_1 \to H_2</math> between Hilbert spaces is the bounded linear operator <math>T^* : H_2 \to H_1</math> such that <math>\langle Tx, y \rangle = \langle x, T^* y \rangle</math> for each <math>x \in H_1, y \in H_2</math>.}}

{{term|1=approximate identity}}
{{defn|1=In a not-necessarily-unital Banach algebra, an [[approximate identity]] is a sequence or a net <math>\{ u_i \}</math> of elements such that <math>u_i x \to x, x u_i \to x</math> as <math>i \to \infty</math> for each ''x'' in the algebra.}}

{{term|1=approximation property}}
{{defn|1=A Banach space is said to have the [[approximation property]] if every compact operator is a limit of finite-rank operators.}}

{{glossary end}}

==B==
{{glossary}}

{{term|1=Baire}}
{{defn|1=The [[Baire category theorem]] states that a [[complete metric space]] is a [[Baire space]]; if <math>U_i</math> is a sequence of open dense subsets, then <math>\cap_1^{\infty} U_i</math> is dense.}}

{{term|1=Banach}}
{{defn|no=1|1=A [[Banach space]] is a normed vector space that is complete as a metric space.}}
{{defn|no=2|1=A [[Banach algebra]] is a Banach space that has a structure of a possibly non-unital [[associative algebra]] such that
:<math>\|x y \| \le \|x\| \|y\|</math> for every <math>x, y</math> in the algebra.}}
{{defn|no=3|1=A [[Banach disc]] is a continuous linear image of a unit ball in a Banach space.}}

{{term|1=balanced}}
{{defn|1=A subset ''S'' of a vector space over real or complex numbers is [[balanced subset|balanced]] if <math>\lambda S \subset S</math> for every scalar <math>\lambda</math> of length at most one.}}

{{term|1=barrel}}
{{defn|no=1|A [[barrel]] in a topological vector space is a subset that is closed, convex, balanced and absorbing.}}
{{defn|no=2|A topological vector space is [[barrelled space|barrelled]] if every barrell is a neighborhood of zero (that is, contains an open neighborhood of zero).}}

{{term|Bessel}}
{{defn|[[Bessel's inequality]] states: given an orthonormal set ''S'' and a vector ''x'' in a Hilbert space,
:<math>\sum_{u \in S} |\langle x, u \rangle|^2 \le \|x\|^2</math>,<ref name="sum convention" />
where the equality holds if and only if ''S'' is an orthonormal basis; i.e., maximal orthonormal set.}}

{{term|bipolar}}
{{defn|1=[[bipolar theorem]].}}

{{term|1=bounded}}
{{defn|1=A [[bounded operator]] is a linear operator between Banach spaces for which the image of the unit ball is bounded.}}

{{term|1=bornological}}
{{defn|1=A [[bornological space]].}}

{{term|1=Birkhoff orthogonality}}
{{defn|1=Two vectors  ''x'' and ''y'' in a [[normed linear space]] are said to be '''Birkhoff orthogonal''' if <math>\| x + \lambda y \| \ge \|x\|</math> for all scalars λ. If the normed linear space is a Hilbert space, then it is equivalent to the usual orthogonality.}}

{{term|Borel}}
{{defn|1=[[Borel functional calculus]]}}

{{glossary end}}

==C==
{{glossary}}

{{term|1=c}}
{{defn|1=[[c space]].}}

{{term|1=Calkin}}
{{defn|1=The [[Calkin algebra]] on a Hilbert space is the quotient of the algebra of all bounded operators on the Hilbert space by the ideal generated by compact operators.}}

{{term|1=Cauchy–Schwarz inequality}}
{{defn|1=The [[Cauchy–Schwarz inequality]] states: for each pair of vectors <math>x, y</math> in an inner-product space,
:<math>|\langle x, y \rangle| \le \|x\| \|y\|</math>.}}

{{term|1=closed}}
{{defn|no=1|1=The [[closed graph theorem]] states that a linear operator between Banach spaces is continuous (bounded) if and only if it has closed graph.}}
{{defn|no=2|1=A [[closed operator]] is a linear operator whose graph is closed.}}
{{defn|no=3|1=The [[closed range theorem]] says that a densely defined closed operator has closed image (range) if and only if the transpose of it has closed image.}}

{{term|1=commutant}}
{{defn|no=1|1=Another name for "[[centralizer (ring theory)|centralizer]]"; i.e., the commutant of a subset ''S'' of an algebra is the algebra of the elements commuting with each element of ''S'' and is denoted by <math>S'</math>.}}
{{defn|no=2|1=The [[von Neumann double commutant theorem]] states that a nondegenerate *-algebra <math>\mathfrak{M}</math> of operators on a Hilbert space is a von Neumann algebra if and only if <math>\mathfrak{M}'' = \mathfrak{M}</math>.}}

{{term|1=compact}}
{{defn|1=A [[compact operator]] is a linear operator between Banach spaces for which the image of the unit ball is precompact.}}

{{term|1=Connes}}
{{defn|1=[[Connes fusion]].}}

{{term|1=C*}}
{{defn|1=A [[C* algebra]] is an involutive Banach algebra satisfying <math>\|x^* x\| = \|x^*\| \|x\|</math>.}}

{{term|1=convex}}
{{defn|1=A [[locally convex space]] is a topological vector space whose topology is generated by convex subsets.}}

{{term|1=cyclic}}
{{defn|1=Given a representation <math>(\pi, V)</math> of a Banach algebra <math>A</math>, a [[cyclic vector]] is a vector <math>v \in V</math> such that <math>\pi(A)v</math> is dense in <math>V</math>.}}

{{glossary end}}

==D==
{{glossary}}

{{term|1=dilation}}
{{defn|1=[[dilation (operator theory)]].}}

{{term|1=direct}}
{{defn|1=Philosophically, a [[direct integral]] is a continuous analog of a direct sum.}}

{{term|1=Douglas}}
{{defn|1=[[Douglas' lemma]]}}

{{term|Dunford}}
{{defn|1=[[Dunford–Schwartz theorem]]}}

{{term|1=dual}}
{{defn|no=1|The [[continuous dual]] of a topological vector space is the vector space of all the continuous linear functionals on the space.}}
{{defn|no=2|The [[algebraic dual]] of a topological vector space is the dual vector space of the underlying vector space.}}

{{glossary end}}

==E==
{{glossary}}

{{term|1=Eidelheit}}
{{defn|1=[[A theorem of Eidelheit]].}}

{{glossary end}}

==F==
{{glossary}}

{{term|1=factor}}
{{defn|1=A [[factor (functional analysis)|factor]] is a von Neumann algebra with trivial center.}}

{{term|1=faithful}}
{{defn|1=A linear functional <math>\omega</math> on an involutive algebra is [[faithful linear functional|faithful]] if <math>\omega(x^*x) \ne 0</math> for each nonzero element <math>x</math> in the algebra.}}

{{term|1=Fréchet}}
{{defn|1=A [[Fréchet space]] is a topological vector space whose topology is given by a countable family of seminorms (which makes it a metric space) and that is complete as a metric space.}}

{{term|1=Fredholm}}
{{defn|1=A [[Fredholm operator]] is a bounded operator such that it has closed range and the kernels of the operator and the adjoint have finite-dimension.}}

{{glossary end}}

==G==
{{glossary}}

{{term|1=Gelfand}}
{{defn|no=1|1=The [[Gelfand–Mazur theorem]] states that a Banach algebra that is a division ring is the field of complex numbers.}}
{{defn|no=2|1=The [[Gelfand representation]] of a commutative Banach algebra <math>A</math> with spectrum <math>\Omega(A)</math> is the algebra homomorphism <math>F: A \to C_0(\Omega(A))</math>, where <math>C_0(X)</math> denotes the algebra of continuous functions on <math>X</math> vanishing at infinity, that is given by <math>F(x)(\omega) = \omega(x)</math>. It is a *-preserving isometric isomorphism if <math>A</math> is a commutative C*-algebra.}}

{{term|1=Grothendieck}}
{{defn|no=1|1=[[Grothendieck's inequality]].}}
{{defn|no=2|1=[[Grothendieck's factorization theorem]].}}

{{glossary end}}

== H ==
{{glossary}}

{{term|1=Hahn–Banach}}
{{defn|1=The [[Hahn–Banach theorem]] states: given a linear functional <math>\ell</math> on a subspace of a complex vector space ''V'', if the absolute value of <math>\ell</math> is bounded above by a seminorm on ''V'', then it extends to a linear functional on ''V'' still bounded by the seminorm. Geometrically, it is a generalization of the [[hyperplane separation theorem]].}}

{{term|1=Heine}}
{{defn|1=A topological vector space is said to have the [[Heine–Borel property]] if every closed and bounded subset is compact. Riesz's lemma says a Banach space with the Heine–Borel property must be finite-dimensional.}}

{{term|1=Hilbert}}
{{defn|no=1|1=A [[Hilbert space]] is an inner product space that is complete as a metric space.}}
{{defn|no=2|1=In the [[Tomita–Takesaki theory]], a (left or right) Hilbert algebra is a certain algebra with an involution.}}

{{term|1=Hilbert–Schmidt}}
{{defn|no=1|1=The [[Hilbert–Schmidt norm]] of a bounded operator <math>T</math> on a Hilbert space is <math>\sum_i \|T e_i \|^2</math> where <math>\{ e_i \}</math> is an orthonormal basis of the Hilbert space.}}
{{defn|no=2|1=A [[Hilbert–Schmidt operator]] is a bounded operator with finite Hilbert–Schmidt norm.}}

{{glossary end}}

== I ==
{{glossary}}

{{term|1=index}}
{{defn|no=1|1=The index of a Fredholm operator <math>T : H_1 \to H_2</math> is the integer <math>\operatorname{dim}(\operatorname{ker}(T^*)) - \operatorname{dim}(\operatorname{ker}(T))</math>.}}
{{defn|no=2|The [[Atiyah–Singer index theorem]].}}

{{term|1=index group}}
{{defn|The [[index group]] of a unital Banach algebra is the quotient group <math>G(A)/G_0(A)</math> where <math>G(A)</math> is the unit group of ''A'' and <math>G_0(A)</math> the identity component of the group.}}

{{term|1=inner product}}
{{defn|no=1|1=An [[inner product]] on a real or complex vector space <math>V</math> is a function <math>\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}</math> such that for each <math>v, w \in V</math>, (1) <math>x \mapsto \langle x, v \rangle</math> is linear and (2) <math>\langle v, w \rangle = \overline{\langle w, v\rangle}</math> where the bar means complex conjugate.}}
{{defn|no=2|1=An [[inner product space]] is a vector space equipped with an inner product.}}

{{term|1=involution}}
{{defn|no=1|1=An [[Involution (mathematics)|involution]] of a Banach algebra ''A'' is an isometric endomorphism <math>A \to A, \, x \mapsto x^*</math> that is conjugate-linear and such that <math>(xy)^* = (yx)^*</math>.}}
{{defn|no=2|1=An [[involutive Banach algebra]] is a Banach algebra equipped with an involution.}}

{{term|1=isometry}}
{{defn|1=A [[linear isometry]] between normed vector spaces is a linear map preserving norm.}}

{{glossary end}}

==K==
{{glossary}}

{{term|1=Köthe}}
{{defn|1=A [[Köthe sequence space]]. For now, see https://mathoverflow.net/questions/361048/on-k%C3%B6the-sequence-spaces}}

{{term|1=Krein–Milman}}
{{defn|1=The [[Krein–Milman theorem]] states: a nonempty compact convex subset of a locally convex space has an extremal point.}}

{{term|1=Krein–Smulian}}
{{defn|1=[[Krein–Smulian theorem]]}}

{{glossary end}}

==L==
{{glossary}}

{{term|1=Linear}}
{{defn|1=[[Linear Operators]] is a three-value book by Dunford and Schwartz.}}

{{term|1=Locally convex algebra}}
{{defn|1=A [[locally convex algebra]] is an algebra whose underlying vector space is a locally convex space and whose multiplication is continuous with respect to the locally convex space topology.}}

{{glossary end}}

==M==
{{glossary}}

{{term|Mazur}}
{{defn|[[Mazur–Ulam theorem]].}}

{{term|Montel}}
{{defn|1=[[Montel space]].}}

{{glossary end}}

==N==
{{glossary}}

{{term|1=nondegenerate}}
{{defn|1=A representation <math>(\pi, V)</math> of an algebra <math>A</math> is said to be nondegenerate if for each vector <math>v \in V</math>, there is an element <math>a \in A</math> such that <math>\pi(a) v \ne 0</math>.}}

{{term|1=noncommutative}}
{{defn|no=1|1=[[noncommutative integration]]}}
{{defn|no=2|1=[[noncommutative torus]]}}

{{term|1=norm}}
{{defn|no=1|1=A [[norm (mathematics)|norm]] on a vector space ''X'' is a real-valued function <math>\| \cdot \| : X \to \mathbb{R}</math> such that for each scalar <math>a</math> and vectors <math>x, y</math> in <math>X</math>, (1) <math>\| ax\| = |a| \| x \|</math>, (2) (triangular inequality) <math>\| x + y \| \le \| x \| + \| y \|</math> and (3) <math>\| x \| \ge 0</math> where the equality holds only for <math>x = 0</math>.}}
{{defn|no=2|1=A [[normed vector space]] is a real or complex vector space equipped with a norm <math>\| \cdot \|</math>. It is a metric space with the distance function <math>d(x, y) = \| x - y \|</math>.}}

{{term|1=normal}}
{{defn|1=An operator is [[normal operator|normal]] if it and its adjoint commute.}}

{{term|1=nuclear}}
{{defn|1=See [[nuclear operator]].}}

{{glossary end}}

==O==
{{glossary}}

{{term|1=one}}
{{defn|1=A [[one parameter group]] of a unital Banach algebra ''A'' is a continuous group homomorphism from <math>(\mathbb{R}, +)</math> to the unit group of ''A''.}}

{{term|1=open}}
{{defn|1=The [[open mapping theorem (functional analysis)|open mapping theorem]] says a surjective continuous linear operator between Banach spaces is an open mapping.}}

{{term|1=orthonormal}}
{{defn|no=1|1=A subset ''S'' of a Hilbert space is [[orthonormal]] if, for each ''u'', ''v'' in the set, <math>\langle u, v \rangle</math> = 0 when <math>u \ne v</math> and <math>= 1</math> when <math>u = v</math>.}}
{{defn|no=2|1=An [[orthonormal basis]] is a maximal orthonormal set (note: it is *not* necessarily a vector space basis.)}}

{{term|1=orthogonal}}
{{defn|no=1|1=Given a Hilbert space ''H'' and a closed subspace ''M'', the [[orthogonal complement]] of ''M'' is the closed subspace <math>M^{\bot} = \{ x \in H | \langle x, y \rangle = 0, y \in M \}</math>.}}
{{defn|no=2|1=In the notations above, the [[orthogonal projection]] <math>P</math> onto ''M'' is a (unique) bounded operator on ''H'' such that <math>P^2 = P, P^* = P, \operatorname{im}(P) = M, \operatorname{ker}(P) = M^{\bot}.</math>}}

{{glossary end}}

==P==
{{glossary}}

{{term|1=Parseval}}
{{defn|1=[[Parseval's identity]] states: given an orthonormal basis ''S'' in a Hilbert space, <math>\| x \|^2 = \sum_{u \in S} |\langle x, u \rangle|^2</math>.<ref name="sum convention">Here, the part of the assertion is <math>\sum_{u \in S} \cdots</math> is well-defined; i.e., when ''S'' is infinite, for countable totally ordered subsets <math>S' \subset S</math>, <math>\sum_{u \in S'} \cdots</math> is independent of <math>S'</math> and <math>\sum_{u \in S} \cdots</math> denotes the common value.</ref>}}

{{term|1=positive}}
{{defn|1=A linear functional <math>\omega</math> on an involutive Banach algebra is said to be [[positive linear functional|positive]] if <math>\omega(x^* x) \ge 0</math> for each element <math>x</math> in the algebra.}}

{{term|1=predual}}
{{defn|1=[[predual]].}}

{{term|1=projection}}
{{defn|1=An operator ''T'' is called a [[projection (functional analysis)|projection]] if it is an idempotent; i.e., <math>T^2 = T</math>.}}

{{glossary end}}

==Q==
{{glossary}}

{{term|1=quasitrace}}
{{defn|1=[[Quasitrace]].}}

{{glossary end}}

==R==
{{glossary}}

{{term|1=Radon}}
{{defn|1=See [[Radon measure]].}}

{{term|1=Riesz decomposition}}
{{defn|1=[[Riesz decomposition]].}}

{{term|1=Riesz's lemma}}
{{defn|1=[[Riesz's lemma]].}}

{{term|1=reflexive}}
{{defn|1=A [[reflexive space]] is a topological vector space such that the natural map from the vector space to the second (topological) dual is an isomorphism.}}

{{term|1=resolvent}}
{{defn|1=The [[Resolvent set|resolvent]] of an element ''x'' of a unital Banach algebra is the complement in <math>\mathbb{C}</math> of the spectrum of ''x''.}}

{{term|1=Ryll-Nardzewski}}
{{defn|1=[[Ryll-Nardzewski fixed-point theorem]].}}

{{glossary end}}

==S==
{{glossary}}

{{term|1=Schauder}}
{{defn|[[Schauder basis]].}}

{{term|Schatten}}
{{defn|[[Schatten class]]}}

{{term|1=selection}}
{{defn|1=[[Michael selection theorem]].}}

{{term|1=self-adjoint}}
{{defn|1=A [[self-adjoint operator]] is a bounded operator whose adjoint is itself.}}

{{term|1=separable}}
{{defn|1=A [[separable Hilbert space]] is a Hilbert space admitting a finite or countable orthonormal basis.}}

{{term|1=spectrum}}
{{defn|no=1|1=The spectrum of an element ''x'' of a unital Banach algebra is the set of complex numbers <math>\lambda</math> such that <math>x - \lambda</math> is not invertible.}}
{{defn|no=2|1=The [[spectrum of a commutative Banach algebra]] is the set of all characters (a homomorphism to <math>\mathbb{C}</math>) on the algebra.}}

{{term|1=spectral}}
{{defn|no=1|1=The [[spectral radius]] of an element ''x'' of a unital Banach algebra is <math display="inline">\sup_{\lambda} |\lambda|</math> where the sup is over the spectrum of ''x''.}}
{{defn|no=2|1=The [[spectral mapping theorem]] states: if ''x'' is an element of a unital Banach algebra and ''f'' is a holomorphic function in a neighborhood of the spectrum <math>\sigma(x)</math> of ''x'', then <math>f(\sigma(x)) = \sigma(f(x))</math>, where <math>f(x)</math> is an element of the Banach algebra defined via the [[Cauchy's integral formula]].}}

{{term|1=state}}
{{defn|1=A [[state (functional analysis)|state]] is a positive linear functional of norm one.}}

{{term|1=symmetric}}
{{defn|A linear operator ''T'' on a pre-Hilbert space is [[symmetric operator|symmetric]] if <math>(Tx, y) = (x, Ty).</math>}}

{{glossary end}}

==T==
{{glossary}}

{{term|1=tensor product}}
{{defn|no=1|1=See [[topological tensor product]]. Note it is still somewhat of an open problem to define or work out a correct tensor product of topological vector spaces, including Banach spaces.}}
{{defn|no=2|A [[projective tensor product]].}}

{{term|1=topological}}
{{defn|no=1|1=A [[topological vector space]] is a vector space equipped with a [[topology (mathematics)|topology]] such that (1) the topology is [[Hausdorff topology|Hausdorff]] and (2) the addition <math>(x, y) \mapsto x + y</math> as well as scalar multiplication <math>(\lambda, x) \mapsto \lambda x</math> are continuous.}}
{{defn|no=2|1=A linear map <math>f: E \to F</math> is called a [[topological homomorphism]] if <math>f : E \to \operatorname{im}(f)</math> is an open mapping.}}
{{defn|no=3|1=A sequence <math>\cdots \to E_{n -1} \to E_n \to E_{n+1} \to \cdots</math> is called [[topologically exact]] if it is an [[exact sequence]] on the underlying vector spaces and, moreover, each <math> E_n \to E_{n+1}</math> is a topological homomorphism.}}

{{glossary end}}

==U==
{{glossary}}

{{term|1=ultraweak}}
{{defn|1=[[ultraweak topology]].}}

{{term|1=unbounded operator}}
{{defn|1=An [[unbounded operator]] is a partially defined linear operator, usually defined on a dense subspace.}}

{{term|1=uniform boundedness principle}}
{{defn|1=The [[uniform boundedness principle]] states: given a set of operators between Banach spaces, if <math display="inline">\sup_T |Tx| < \infty</math>, sup over the set, for each ''x'' in the Banach space, then <math display="inline">\sup_T \|T\| < \infty</math>.}}

{{term|1=unitary}}
{{defn|no=1|1=A [[unitary operator]] between Hilbert spaces is an invertible bounded linear operator such that the inverse is the adjoint of the operator.}}
{{defn|no=2|1=Two representations <math>(\pi_1, H_1), (\pi_2, H_2)</math> of an involutive Banach algebra ''A'' on Hilbert spaces <math>H_1, H_2</math> are said to be [[unitary representation|unitarily equivalent]] if there is a unitary operator <math>U: H_1 \to H_2</math> such that <math>\pi_2(x) U = U \pi_1(x)</math> for each ''x'' in ''A''.}}

{{glossary end}}

==V==
{{glossary}}

{{term|1=von Neumann}}
{{defn|no=1|1=A [[von Neumann algebra]].}}
{{defn|no=2|1=[[von Neumann's theorem]].}}
{{defn|no=3|1=[[Von Neumann's inequality]].}}

{{glossary end}}

==W==
{{glossary}}

{{term|1=W*}}
{{defn|1=A W*-algebra is a C*-algebra that admits a faithful representation on a Hilbert space such that the image of the representation is a von Neumann algebra.}}

{{glossary end}}

== References ==
{{Reflist}}
* Bourbaki, ''Espaces vectoriels topologiques''
* {{Citation | last1=Connes | first1=Alain | author1-link=Alain Connes | title=Non-commutative geometry | url=https://archive.org/details/noncommutativege0000conn | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-185860-5 | year=1994 | url-access=registration }}
* {{Conway A Course in Functional Analysis|edition=2}} <!--{{sfn|Conway|1990|p=}}-->
* {{Dunford Schwartz Linear Operators Part 1 General Theory}}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->
* M. Takesaki, ''Theory of Operator Algebras I'', Springer, 2001, 2nd printing of the first edition 1979.
* {{citation | last1=Yoshida| first1=Kôsaku | title=Functional Analysis | year=1980| publisher=Springer |edition=sixth}}

== Further reading ==
*Antony Wassermann's lecture notes at http://iml.univ-mrs.fr/~wasserm/
*Jacob Lurie's lecture notes on a von Neumann algebra at https://www.math.ias.edu/~lurie/261y.html
* https://mathoverflow.net/questions/408415/takesaki-theorem-2-6

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[[Category:Functional analysis]]
[[Category:Glossaries of mathematics|Functional analysis]]