Sublinear functionType of function in linear algebraIn linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm, on a vector space is a real-valued function with some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous.
Minkowski functionalIn mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
Hahn–Banach theoremTheorem on extension of bounded linear functionalsIn functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some normed vector space to the whole space. In a different guise the theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem and has numerous uses in convex geometry.Norm (mathematics)Length in a vector spaceIn mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
Absorbing setIn functional analysis and related areas of mathematics an absorbing set in a vector space is a setS{\displaystyle S} which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset.
Absolutely convex setIn mathematics, a subsetC of a real or complexvector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.Topological vector spaceIn mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions.Balanced setIn linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field K {\displaystyle \mathbb {K} } with an absolute value function | ⋅ | {\displaystyle |\cdot |} ) is a set S {\displaystyle S} such that a S ⊆ S {\displaystyle aS\subseteq S} for all scalars a {\displaystyle a} satisfying | a | ≤ 1.
Convex setIn geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube is a convex set, but anything that is hollow or has an indent, such as a crescent shape, is not convex. The boundary of a convex set in the plane is always a convex curve.
Functional analysisFunctional 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 (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Positive definitenessIn mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite.*As an Amazon Associate I earn from qualifying purchases.