Locally convex topological vector space

Locally convex spaces (more precisely, locally convex topological vector spaces ) are in the mathematical branch of functional analysis examined topological vector spaces with additional properties. It involves topological vector spaces, in which each point has " arbitrarily small " convex environments. Alternatively locally convex spaces can also be defined as vector spaces whose topology is generated by a family of semi-norms.

A locally convex space can be viewed as a generalization of a normed vector space or a normalizable vector space, because the standard balls around 0 are convex environments of the zero point.

Geometric definition

A topological vector space ( over the field of real numbers or the field of complex numbers ) is called locally convex if every neighborhood of zero (ie neighborhood of the origin ) contains an open subset with the following three properties:

• Is convex.
• Is absorbing.
• Is balanced.

A subset of a real or complex vector space is called while absorbing if there exists for every vector in a strictly positive real number such that for every real or complex number with an element of.

A subset of a real or complex vector space is called balanced if for each vector with the vector lies in, and each number also. In the case of a real vector space, this means that the distance is from to in; For a complex vector space, it means that the " disc " includes. Due to this geometric meaning, such amounts are sometimes called circular.

A balanced and convex set is called absolutely convex set.

It turns out that it is possible to dispense with the second and third condition. There are exactly then a base of neighborhoods of convex, absorbent and balanced amounts, if there is a base of neighborhoods of convex sets. Two such environment bases must of course do not match, but the existence of one implies the existence of the other.

Defined by seminorms

Locally convex spaces can be characterized by semi-norms systems: A topological vector space is called locally convex if its topology is defined by a family of semi-norms. That is, a network converges if and only if it converges with respect to all seminorms of; more precisely: It is if and only if for all semi-norms. The balls, wherein, thereby forming a sub-base of the topology, the amounts are absolutely convex environments.

Conversely, if a neighborhood base of absolutely convex sets is given, the corresponding Minkowski functionals form a defining seminorms system.

Examples

Properties

Meets the seminorm amount from the above definition, the space is a Hausdorff space. Many authors consider only Hausdorff'sche locally convex spaces.

Hausdorff, locally convex spaces have a sufficient number of continuous, linear functionals to separate points, that is, for all there is a steady, linear functional with. This is reflected in the validity of important phrases like

• The Hahn- Banach
• The separation theorem
• The Krein - Milman

The continuous linear functionals on a topological vector space V if and only separate the points if there is a coarser topology on V which makes V a Hausdorff, locally convex space. The study of locally convex spaces by means of continuous, linear functionals leads to a very far-reaching theory that is not possible for general topological vector spaces. There are topological vector spaces, in addition to the zero functional have no further continuous linear functional.

Generalizations

• Topological vector space
• Topological group
• Uniform space

Specific classes of locally convex spaces

Many classes of locally convex spaces are characterized by the validity of certain sentences which are known from the theory of Banach spaces or normed spaces, from. Thus, for example, the barreled spaces precisely those locally convex spaces in which the Banach - Steinhaus still applies. These sets can be examined in the appropriate space classes in "pure culture ", its scope is clear. The best known area classes are:

• Bornological spaces and rooms ultrabornologische
• Barreled spaces and quasi- barreled spaces
• Metrizable locally convex spaces
• Fréchet spaces
• Normed vector spaces
• Banach spaces
• Hilbert spaces

Spaces of differentiable or holomorphic functions carry natural locally convex topologies, give their properties to another room classes occasion. The most important room classes that lead to a deeper understanding of the locally convex theory, are about

• Nuclear Facilities
• Montelräume
• Schwartz spaces
• (DF )-spaces and GdF - rooms
• Quasinormierbare rooms
• (LF )-spaces and LB - spaces
• Rooms with tissue

Historical Remarks

In 1906 stelle M. Fréchet found on concerning the pointwise convergence that the " conclusion " of the set of bounded continuous functions on the set of all bounded functions can not be described by the set of all limits of sequences from. The required general area as that term was in 1914 introduced by F. Hausdorff in general topology, found in the functional analysis for the first time by J. von Neumann in his description of the weak and strong neighborhoods of zero in Hilbert space, application, and a generalization to Banach spaces was not attempted. The ambient term for more general situations can be found in S. Banach (1932) and Bourbaki (1938 ) in studies on the weak -* topology, first being restricted to separable spaces so that the unit ball in the dual space is metrizable. Although the study of normed spaces was in the foreground, so it was clear that more general classes of area occur naturally.

S. Banach, S. Mazur and Orlicz W considered spaces whose topology is given by a sequence of semi-norms, and defined the distance. For full under these rooms, which today is called Frécheträume, the sentence could be proved by the closed graph. But that is not sufficient even metrizable spaces, showed the determination of J. von Neumann in 1929 that the weak topology is not metrizable on infinite-dimensional Hilbert spaces.

1934 new types of spaces, so-called sequence spaces introduced and their duality theory developed in the context of sequence spaces of G. Köthe and O. Toeplitz. In this context, the term of the strong dual space appeared.

The sense of the metric in limited quantities did not behave as in the case of normed spaces, it required a topological characterization of the boundedness, as given in the references cited below J. v. Neumann's work from 1935. This boundedness concept can be found in the same year at AN Kolmogorov in the proof of the statement that a topological vector space if and only normalizable if it has a limited neighborhood of zero. Von Neumann's work for the first time contains the general definition of locally convex space, the equivalence of the above geometric definition and the definition of semi-norms is proven there. Thus, the ideas of Köthe and Toeplitz could be carried out in a more general context. Milestones were the results of G. Mackey of 1946, see Theorem of Mackey, Mackey - Arens theorem of, and the studies of tensor products of A. Grothendieck from the year 1953. More importance came to the locally convex spaces by L. Schwartz, A. Grothendieck, SL Sobolev, S. Bochner and other investigations carried out in the theory of partial differential equations and the associated justification of the theory of distributions. The set of core led Grothendieck to the important concept of the nuclear space.