In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points...

functionals. Fréchet spaces are locally convex topological vector spaces that are completely metrizable (with a choice of complete metric). They are generalizations...

Fréchet space be locally convex (discussed below). The topology of every Fréchet space is induced by some translation-invariant complete metric. Conversely...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

manifolds and sub-Riemannian manifolds. Any complete and convex metric space is a length metric space (Khamsi & Kirk 2001, Theorem 2.16), a result of Karl...

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function...

this glossary. A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning...

convex sets and convex functions is called convex analysis. Spaces in which convex sets are defined include the Euclidean spaces, the affine spaces over...

a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is...

points Convex polygon, a polygon which encloses a convex set of points Convex polytope, a polytope with a convex set of points Convex metric space, a generalization...

general locally convex metric spaces. The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected...

Banach space (/ˈbɑː.nʌx/, Polish pronunciation: [ˈba.nax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that...

These definitions coincide for subsets of a complete metric space, but not in general. A metric space ( M , d ) {\displaystyle (M,d)} is totally bounded...

a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended"...

space – Locally convex topological vector space that is also a complete metric space Hilbert space – Type of vector space in math K-space (functional analysis)...

is also a convex metric space with respect to the geodesic distance. A subset of n-dimensional Euclidean space En with its usual flat metric is geodesically...

vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable...

strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is...

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number...

mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional...

pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced...

locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions...

(Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X →...

A space with this metric is called Minkowski space. The hypersurface F 0 {\displaystyle F_{0}} is convex and can be irregular. The defined metric is...

also known as a Helly family. A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1, in the...

\varphi (t)=\min(t,1).} Such a metric is called Lévy-metric for L 0 . {\displaystyle L^{0}.} Under this metric the space L 0 {\displaystyle L^{0}} is complete...

of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental,...

Another misnomer is Minkowski metric, but Minkowski space is not a metric space. The group of transformations for Minkowski space that preserves the spacetime...

aforementioned equivalence of metric functions remains valid if √q(x − y) is replaced with M(x − y), where M is any convex positive homogeneous function...

distance does not form a metric space, as it does not satisfy the triangle inequality. However it is a smooth, strictly convex function of the two points...