In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function. For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from into its bidual is a homeomorphism. A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, the Gelfand representation in functional analysis is either of two things:
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.
In mathematics, a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebraA, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum  is also naturally a topological space; this is similar to the notion of the spectrum of a ring.
In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces, typically of infinite dimension. It is one aspect of nonlinear functional analysis.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem, is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual is bijective. If this map is also an isomorphism of TVSs then it is called reflexive.
In functional analysis, the Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz.
In functional analysis, a topological homomorphism or simply homomorphism is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.
This is a glossary for the terminology in a mathematical field of functional analysis.
F. Riesz's theorem is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.
In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed.
In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open map. As described below, for certain broad categories of topological vector spaces, all surjective linear operators are necessarily almost open.
