Topological vector Space

The topological vector spaces are one of the basic structures of the analyzes functional. They are spaces provided with a topological structure associated with a structure of vector Space.

Known examples of topological vector spaces are the spaces of Banach and the spaces of Hilbert.

Definition

A topological vector space (" e.v.t") E is a vector Space on a topological body K (generally R or C provided with their usual topology) provided with a Topologie compatible with the structure of vector space , i.e. checking the following conditions:
  • the sum of two vectors is a continuous application of E X E in E ,
  • the product of a scalar by a vector is a continuous application of K X E in E .

The category of the topological vector spaces on a topological body K is noted TVSK or TVectK where the objects are the K - topological vector spaces and the Morphisme S are the applications K - linear continuous.

Vicinities of the origin

Absorbing unit

A part \ quad U of a vector space \ quad E on \ mathbb K = \ mathbb R or \ mathbb C is absorbing if:
\ forall v \ in E \ quad \ exists \ alpha \ in \ mathbb R_+^* \ quad \ forall \ lambda \ in K \ quad |\ lambda|\ the \ alpha \ Rightarrow \ lambda v \ in U
; Theorem
; Any vicinity of the origin is an absorbing unit.
Indeed if \ mathcal V is a vicinity of 0 and if v are an unspecified vector, it results from the continuity of the application (partial) of \ mathbb K in \ mathbb E: \ lambda \ mapsto \ lambda v that there exists a vicinity of 0 in \ mathbb K which one can restrict with |\ lambda|\ the \ alpha whose image is in \ mathcal V and thus |\ lambda|\ the \ alpha \ Rightarrow \ lambda v \ in \ mathcal V.

Symmetrical unit

A part \ quad U of a e.v.t \ quad E on \ mathbb K = \ mathbb R or \ mathbb C is symmetrical if:
\ forall v \ in U \ quad - v \ in U.

Balanced together

A part \ quad U of a e.v.t \ quad E on \ mathbb K = \ mathbb R or \ mathbb C is balanced if:
\ forall \ lambda \ in K \ quad \ forall v \ in U \ quad |\ lambda|\ the 1 \ Rightarrow \ lambda v \ in U

Balanced core of part of E containing the origin

The core balanced NR of a part has E containing 0 is the meeting of the balanced parts of E included in A. This core is nonempty since {0} is a balanced part included in A. It is a balanced unit because any meeting of balanced units is balanced (since if X \ in N X belongs to a balanced part included in NR). NR is thus the greatest balanced unit included in has

; Theorem

; That is to say N the balanced core of a A unit containing the origin. So that \ quad v \ in N, it is necessary and it is enough that for any scalar \ lambda \, checking |\ lambda|\ the 1 one has \ quad \ lambda Indeed v \ in A.
if v \ in NR then for any scalar \ quad \ lambda checking |\ lambda|\ the 1 one has \ lambda v \ in NR \ in A.
Réciproquement if v check the condition |\ lambda|\ the 1 \ Rightarrow \ lambda v \ in A, let us suppose that v \ not \ in N. By posing N'=N \ cup \ {\ driven v/ |\ driven|\ the 1 \} one sees that is not a unit balanced included in has and containing NR strictly, which is contradictory.

Types of topological vector spaces

According to the application that one makes some, one generally uses additional constraints on the topological structure of space. Below some particular types of topological spaces, about classified according to their “kindness are”.

  • topological vector Spaces locally convex: in these spaces, any point admits a base of convex vicinities. By the technique known under the name of Functional of Minkowski , one can show that a space is locally convex if and only if its topology can be defined by a family of Semi-norme S. local convexity is the minimum necessary for geometrical arguments like the Théorème of Hahn-Banach.

  • Spaces arbors: spaces locally convex where the Théorème of Banach-Steinhaus applies.
  • Spaces of Montel: spaces arbors where all Fermé Borné is compact.
  • bornologic Spaces: spaces locally convex where the continuous linear operators with values in a space locally convex are exactly the limited linear operators.
  • Spaces LF
  • Spaces F
  • Spaces of Fréchet
  • nuclear Spaces
  • normalized vector Spaces and pseudo norms: spaces locally convex where topology can be described by single a standard or Semi-norme. In the normalized vector spaces, a linear operator is continuous if and only if it is limited.
  • Spaces of Banach: normalized vector spaces complete. Most of the functional analysis is formulated for spaces of Banach.
  • reflexive Spaces: isomorphous spaces of Banach to their dual double. An important example of reflexive space not is L 1, whose dual one is L ∞ but is strictly contained in the dual one of L ∞.
  • Spaces of Hilbert: they have a scalar Produit; although these spaces can be of infinite size, the majority of the familiar geometrical reasoning in finished dimension also apply.
  • Euclidean Spaces: those are spaces of Hilbert of finished size.

References

  • Alexander Grothendieck, Topological vector spaces , Gordon and Breach Publishers Science, New York, 1973

  • G Köthe, Topological vector spaces , Grundlehren DER mathematischen Wissenschaften, Band 159, Springer-Verlag, New York, 1969
  • Helmuth H. Schaefer, Topological vector spaces. , Springer-Verlag, New York, 1971
  • F Trier, Topological Vector Spaces, Distributions, and Kernels , Academic Near, 1967

See too

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