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.

See also: List of Banach spaces.

- *
- *-homomorphism between involutive Banach algebras is an algebra homomorphism preserving *.

- 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 between Hilbert spaces is the bounded linear operator such that for each .
- approximate identity
- In a not-necessarily-unital Banach algebra, an approximate identity is a sequence or a net of elements such that as 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.

- Baire
- The Baire category theorem states that a complete metric space is a Baire space; if is a sequence of open dense subsets, then 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
- for every in the algebra.

- ,
^{[1]}

- 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 in an inner-product space,
- .

- direct
- Philosophically, a direct integral is a continuous analog of a direct sum.

- factor
- A factor is a von Neumann algebra with trivial center.
- faithful
- A linear functional on an involutive algebra is faithful if for each nonzero element 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.

- 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 with spectrum is the algebra homomorphism , where denotes the algebra of continuous functions on vanishing at infinity, that is given by . It is a *-preserving isometric isomorphism if is a commutative C*-algebra.
- Grothendieck
- Grothendieck's inequality.

- Hahn–Banach
- The Hahn–Banach theorem states: given a linear functional on a subspace of a complex vector space
*V*, if the absolute value of 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 on a Hilbert space is where is an orthonormal basis of the Hilbert space.
- 2. A Hilbert–Schmidt operator is a bounded operator with finite Hilbert–Schmidt norm.

- index
- 1. The index of a Fredholm operator is the integer .
- 2. The Atiyah–Singer index theorem.
- index group
- The index group of a unital Banach algebra is the quotient group where is the unit group of
*A*and the identity component of the group. - inner product
- 1. An inner product on a real or complex vector space is a function such that for each , (1) is linear and (2) 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 that is conjugate-linear and such that . - 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.

- Krein–Milman
- The Krein–Milman theorem states: a nonempty compact convex subset of a locally convex space has an extremal point.

- 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.

- nondegenerate
- A representation of an algebra is said to be nondegenerate if for each vector , there is an element such that .
- noncommutative
- 1. noncommutative integration
- 2. noncommutative torus
- norm
- 1. A norm on a vector space
*X*is a real-valued function such that for each scalar and vectors in , (1) , (2) (triangular inequality) and (3) where the equality holds only for . - 2. A normed vector space is a real or complex vector space equipped with a norm . It is a metric space with the distance function .
- nuclear
- See nuclear operator.

- one
- A one parameter group of a unital Banach algebra
*A*is a continuous group homomorphism from 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, = 0 when and when . - 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 . - 2. In the notations above, the orthogonal projection onto
*M*is a (unique) bounded operator on*H*such that

- Parseval
- Parseval's identity states: given an orthonormal basis
*S*in a Hilbert space, .^{[1]} - positive
- A linear functional on an involutive Banach algebra is said to be positive if for each element in the algebra.

- quasitrace
- Quasitrace.

- 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 of the spectrum of*x*.

- 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 such that is not invertible. - 2. The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to ) on the algebra.
- spectral
- 1. The spectral radius of an element
*x*of a unital Banach algebra is 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 of*x*, then , where 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.

- 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 as well as scalar multiplication are continuous.

- 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 over the set, for each
*x*in the Banach space, then . - 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 of an involutive Banach algebra
*A*on Hilbert spaces are said to be unitarily equivalent if there is a unitary operator such that for each*x*in*A*.

- 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.