** On every infinite-dimensional topological vector space there is a discontinuous linear map.

At this time he was a leading expert in the theory of topological vector spaces.

His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational for Schwartz distributions, and the application of L < sup > p </ sup > spaces in studying linear maps between topological vector spaces.

is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.

Instead, with the topology of compact convergence, C ( a, b ) can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.

This is most often seen in the context of topological vector spaces, but requires only the existence of a continuous " subtraction " operation.

When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.

More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

* If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application defined by is continuous.

Theorem: Let V be a topological vector space

In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces.

To put it more abstractly every semi-normed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm.

All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology ( although the resulting metric spaces need not be the same ).

** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

* Base ( topology ), a topological space

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness — originally called bicompactness — involves families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

Specifically, a topological space is compact if, whenever a collection of open sets covers the space, some subcollection consisting only of finitely many open sets also covers the space.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

A subset K of a topological space X is called compact if it is compact in the induced topology.

Ultimately the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine – Borel compactness in a way that could be applied to the modern notion of a topological space.

* Any finite topological space, including the empty set, is compact.

Then is a compact topological space ; this follows from the Tychonoff theorem.

* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.

# A topological space X is compact.

* A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.

* Permanently singular elements in Banach algebras are topological divisors of zero, i. e., considering extensions B of Banach algebras A some elements that are singular in the given algebra A have a multiplicative inverse element in a Banach algebra extension B. Topological divisors of zero in A are permanently singular in all Banach extension B of A.

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

Kervaire exhibited topological manifolds with no smooth structure at all.

Algebra of continuous functions: a contravariant functor from the category of topological spaces ( with continuous maps as morphisms ) to the category of real associative algebras is given by assigning to every topological space X the algebra C ( X ) of all real-valued continuous functions on that space.

However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies.

Homeomorphisms are the isomorphisms in the category of topological spaces — that is, they are the mappings that preserve all the topological properties of a given space.

The homeomorphisms form an equivalence relation on the class of all topological spaces.

Metrizable spaces inherit all topological properties from metric spaces.

However, in the context of topology, sequences do not fully encode all information about a function between topological spaces.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

or the ( possibly infinite ) Cartesian product of the topological spaces X < sub > i </ sub >, indexed by, and the canonical projections p < sub > i </ sub >: X → X < sub > i </ sub >, the product topology on X is defined to be the coarsest topology ( i. e. the topology with the fewest open sets ) for which all the projections p < sub > i </ sub > are continuous.

The routing table stores only the best possible routes, while link-state or topological databases may store all other information as well.

; Borel algebra: The Borel algebra on a topological space is the smallest-algebra containing all the open sets.

The topological dual space consists of all linear functions from X into the base field K which are continuous with respect to the given topology.

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are " near " S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

More generally yet, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.