title: Glossary of functional analysis
text: # Glossary of functional analysis
  
  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.
  
  * A
  * B
  * C
  * D
  * E
  * F
  * G
  * H
  * I
  * J
  * K
  * L
  * M
  * N
  * O
  * P
  * Q
  * R
  * S
  * T
  * U
  * V
  * W
  * XYZ
  * See also
  * References
  
  See also: List of Banach spaces.
  
      *-homomorphism between involutive Banach algebras is an algebra homomorphism preserving *.
  
  ## A
  
  ### abelian
  
      Synonymous with "commutative"; e.g., an abelian Banach algebra means a commutative Banach algebra.
  
  ### Alaoglu
  
      Alaoglu's theorem states that the closed unit ball in a normed space is compact in the weak-* topology.
  
  ### adjoint
  
      The adjoint of a bounded linear operator $T:H_{1}\to H_{2}$ between Hilbert spaces is the bounded linear operator $T^{*}:H_{2}\to H_{1}$ such that $\langle Tx,y\rangle =\langle x,T^{*}y\rangle$ for each $x\in H_{1},y\in H_{2}$.
  
  ### approximate identity
  
      In a not-necessarily-unital Banach algebra, an approximate identity is a sequence or a net $\{u_{i}\}$ of elements such that $u_{i}x\to x,xu_{i}\to x$ as $i\to \infty$ for each x in the algebra.
  
  ### approximation property
  
      A Banach space is said to have the approximation property if every compact operator is a limit of finite-rank operators.
  
  ## B
  
  ### Baire
  
      The Baire category theorem states that a complete metric space is a Baire space; if $U_{i}$ is a sequence of open dense subsets, then $\cap _{1}^{\infty }U_{i}$ is dense.
  
  ### Banach
  
      1. A Banach space is a normed vector space that is complete as a metric space.
      2. A Banach algebra is a Banach space that has a structure of a possibly non-unital associative algebra such that
         $\|xy\|\leq \|x\|\|y\|$ for every $x,y$ in the algebra.
  
  ### Bessel
  
      Bessel's inequality states: given an orthonormal set S and a vector x in a Hilbert space, where the equality holds if and only if S is an orthonormal basis; i.e., maximal orthonormal set.
         $\sum _{u\in S}|\langle x,u\rangle |^{2}\leq \|x\|^{2}$,
  
  ### bounded
  
      A bounded operator is a linear operator between Banach spaces for which the image of the unit ball is bounded.
  
  ### Birkhoff orthogonality
  
      Two vectors x and y in a normed linear space are said to be Birkhoff orthogonal if $\|x+\lambda y\|\geq \|x\|$ for all scalars λ. If the normed linear space is a Hilbert space, then it is equivalent to the usual orthogonality.
  
  ## C
  
  ### Calkin
  
      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.
  
  ### Cauchy–Schwarz inequality
  
      The Cauchy–Schwarz inequality states: for each pair of vectors $x,y$ in an inner-product space,
         $|\langle x,y\rangle |\leq \|x\|\|y\|$.
  
  ### closed
  
      The closed graph theorem states that a linear operator between Banach spaces is continuous (bounded) if and only if it has closed graph.
  
  ### commutant
  
      1. Another name for "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 $S'$.
      2. The von Neumann double commutant theorem states that a nondegenerate *-algebra ${\mathfrak {M}}$ of operators on a Hilbert space is a von Neumann algebra if and only if ${\mathfrak {M}}''={\mathfrak {M}}$.
  
  ### compact
  
      A compact operator is a linear operator between Banach spaces for which the image of the unit ball is precompact.
  
  ### C*
  
      A C* algebra is an involutive Banach algebra satisfying $\|x^{*}x\|=\|x^{*}\|\|x\|$.
  
  ### convex
  
      A locally convex space is a topological vector space whose topology is generated by convex subsets.
  
  ### cyclic
  
      Given a representation $(\pi ,V)$ of a Banach algebra $A$, a cyclic vector is a vector $v\in V$ such that $\pi (A)v$ is dense in $V$.
  
  ## D
  
  ### direct
  
      Philosophically, a direct integral is a continuous analog of a direct sum.
  
  ## F
  
  ### factor
  
      A factor is a von Neumann algebra with trivial center.
  
  ### faithful
  
      A linear functional $\omega$ on an involutive algebra is faithful if $\omega (x^{*}x)\neq 0$ for each nonzero element $x$ in the algebra.
  
  ### Fréchet
  
      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.
  
  ### Fredholm
  
      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.
  
  ## G
  
  ### Gelfand
  
      1. The Gelfand–Mazur theorem states that a Banach algebra that is a division ring is the field of complex numbers.
      2. The Gelfand representation of a commutative Banach algebra $A$ with spectrum $\Omega (A)$ is the algebra homomorphism $F:A\to C_{0}(\Omega (A))$, where $C_{0}(X)$ denotes the algebra of continuous functions on $X$ vanishing at infinity, that is given by $F(x)(\omega )=\omega (x)$. It is a *-preserving isometric isomorphism if $A$ is a commutative C*-algebra.
  
  ### Grothendieck
  
      Grothendieck's inequality.
  
  ## H
  
  ### Hahn–Banach
  
      The Hahn–Banach theorem states: given a linear functional $\ell$ on a subspace of a complex vector space V, if the absolute value of $\ell$ 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.
  
  ### Hilbert
  
      1. A Hilbert space is an inner product space that is complete as a metric space.
      2. In the Tomita–Takesaki theory, a (left or right) Hilbert algebra is a certain algebra with an involution.
  
  ### Hilbert–Schmidt
  
      1. The Hilbert–Schmidt norm of a bounded operator $T$ on a Hilbert space is $\sum _{i}\|Te_{i}\|^{2}$ where $\{e_{i}\}$ is an orthonormal basis of the Hilbert space.
      2. A Hilbert–Schmidt operator is a bounded operator with finite Hilbert–Schmidt norm.
  
  ## I
  
  ### index
  
      1. The index of a Fredholm operator $T:H_{1}\to H_{2}$ is the integer $\operatorname {dim} (\operatorname {ker} (T^{*}))-\operatorname {dim} (\operatorname {ker} (T))$.
      2. The Atiyah–Singer index theorem.
  
  ### index group
  
      The index group of a unital Banach algebra is the quotient group $G(A)/G_{0}(A)$ where $G(A)$ is the unit group of A and $G_{0}(A)$ the identity component of the group.
  
  ### inner product
  
      1. An inner product on a real or complex vector space $V$ is a function $\langle \cdot ,\cdot \rangle :V\times V\to \mathbb {R}$ such that for each $v,w\in V$, (1) $x\mapsto \langle x,v\rangle$ is linear and (2) $\langle v,w\rangle ={\overline {\langle w,v\rangle }}$ where the bar means complex conjugate.
      2. An inner product space is a vector space equipped with an inner product.
  
  ### involution
  
      1. An involution of a Banach algebra A is an isometric endomorphism $A\to A,\,x\mapsto x^{*}$ that is conjugate-linear and such that $(xy)^{*}=(yx)^{*}$.
      2. An involutive Banach algebra is a Banach algebra equipped with an involution.
  
  ### isometry
  
      A linear isometry between normed vector spaces is a linear map preserving norm.
  
  ## K
  
  ### Krein–Milman
  
      The Krein–Milman theorem states: a nonempty compact convex subset of a locally convex space has an extremal point.
  
  ## L
  
  ### Locally convex algebra
  
      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.
  
  ## N
  
  ### nondegenerate
  
      A representation $(\pi ,V)$ of an algebra $A$ is said to be nondegenerate if for each vector $v\in V$, there is an element $a\in A$ such that $\pi (a)v\neq 0$.
  
  ### noncommutative
  
      1. noncommutative integration
      2. noncommutative torus
  
  ### norm
  
      1. A norm on a vector space X is a real-valued function $\|\cdot \|:X\to \mathbb {R}$ such that for each scalar $a$ and vectors $x,y$ in $X$, (1) $\|ax\|=|a|\|x\|$, (2) (triangular inequality) $\|x+y\|\leq \|x\|+\|y\|$ and (3) $\|x\|\geq 0$ where the equality holds only for $x=0$.
      2. A normed vector space is a real or complex vector space equipped with a norm $\|\cdot \|$. It is a metric space with the distance function $d(x,y)=\|x-y\|$.
  
  ### nuclear
  
      See nuclear operator.
  
  ## O
  
  ### one
  
      A one parameter group of a unital Banach algebra A is a continuous group homomorphism from $(\mathbb {R} ,+)$ to the unit group of A.
  
  ### orthonormal
  
      1. A subset S of a Hilbert space is orthonormal if, for each u, v in the set, $\langle u,v\rangle$ = 0 when $u\neq v$ and $=1$ when $u=v$.
      2. An orthonormal basis is a maximal orthonormal set (note: it is *not* necessarily a vector space basis.)
  
  ### orthogonal
  
      1. Given a Hilbert space H and a closed subspace M, the orthogonal complement of M is the closed subspace $M^{\bot }=\{x\in H|\langle x,y\rangle =0,y\in M\}$.
      2. In the notations above, the orthogonal projection $P$ onto M is a (unique) bounded operator on H such that $P^{2}=P,P^{*}=P,\operatorname {im} (P)=M,\operatorname {ker} (P)=M^{\bot }.$
  
  ## P
  
  ### Parseval
  
      Parseval's identity states: given an orthonormal basis S in a Hilbert space, $\|x\|^{2}=\sum _{u\in S}|\langle x,u\rangle |^{2}$.
  
  ### positive
  
      A linear functional $\omega$ on an involutive Banach algebra is said to be positive if $\omega (x^{*}x)\geq 0$ for each element $x$ in the algebra.
  
  ## Q
  
  ### quasitrace
  
      Quasitrace.
  
  ## R
  
  ### Radon
  
      See Radon measure.
  
  ### Riesz decomposition
  
      Riesz decomposition.
  
  ### Riesz's lemma
  
      Riesz's lemma.
  
  ### reflexive
  
      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.
  
  ### resolvent
  
      The resolvent of an element x of a unital Banach algebra is the complement in $\mathbb {C}$ of the spectrum of x.
  
  ## S
  
  ### self-adjoint
  
      A self-adjoint operator is a bounded operator whose adjoint is itself.
  
  ### separable
  
      A separable Hilbert space is a Hilbert space admitting a finite or countable orthonormal basis.
  
  ### spectrum
  
      1. The spectrum of an element x of a unital Banach algebra is the set of complex numbers $\lambda$ such that $x-\lambda$ is not invertible.
      2. The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to $\mathbb {C}$) on the algebra.
  
  ### spectral
  
      1. The spectral radius of an element x of a unital Banach algebra is $sup_λ|λ|{\textstyle$ \sup _{\lambda }|\lambda |} where the sup is over the spectrum of x.
      2. 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 $\sigma (x)$ of x, then $f(\sigma (x))=\sigma (f(x))$, where $f(x)$ is an element of the Banach algebra defined via the Cauchy's integral formula.
  
  ### state
  
      A state is a positive linear functional of norm one.
  
  ## T
  
  ### tensor product
  
      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.
  
  ### topological
  
      A topological vector space is a vector space equipped with a topology such that (1) the topology is Hausdorff and (2) the addition $(x,y)\mapsto x+y$ as well as scalar multiplication $(\lambda ,x)\mapsto \lambda x$ are continuous.
  
  ## U
  
  ### unbounded operator
  
      An unbounded operator is a partially defined linear operator, usually defined on a dense subspace.
  
  ### uniform boundedness principle
  
      The uniform boundedness principle states: given a set of operators between Banach spaces, if $sup_T|Tx|<∞{\textstyle$ \sup _{T}|Tx|<\infty }, sup over the set, for each x in the Banach space, then $sup_T‖T‖<∞{\textstyle$ \sup _{T}\|T\|<\infty }.
  
  ### unitary
  
      1. A unitary operator between Hilbert spaces is an invertible bounded linear operator such that the inverse is the adjoint of the operator.
      2. Two representations $(\pi _{1},H_{1}),(\pi _{2},H_{2})$ of an involutive Banach algebra A on Hilbert spaces $H_{1},H_{2}$ are said to be unitarily equivalent if there is a unitary operator $U:H_{1}\to H_{2}$ such that $\pi _{2}(x)U=U\pi _{1}(x)$ for each x in A.
  
  ## W
  
  ### W*
  
      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.
  