Flashcards: Functional analysis

An operator is a mapping from one vector space or module to another.

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

In mathematics a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space.

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g); it produces a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

In mathematics, a metric space is a set for which distances between all members of the set are defined.

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

In mathematics, the Hahn–Banach Theorem is a central tool in functional analysis.

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis.

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet.

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map L : V → W between two vector spaces V and W, is the set of all elements v of V for which L(v) = 0, where 0 denotes the zero vector in W.

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G.

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers.

In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.

In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary.

In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-algebras, this representation is an isometric isomorphism.

In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space.