nLab
topological vector space

Context

Functional analysis

Topology

Contents

Idea

A topological vector space, or TVS for short, is a vector space XX over a topological field (usually a local field, more often than not the field of real numbers or the field of complex numbers with the usual topology) kk (called the ground field) equipped with a topology for which the addition and scalar multiplication maps

+:X×XX,:k×XX+: X \times X \to X, \qquad \cdot: k \times X \to X

are continuous.

Much as a topological group is a group object in Top, so a TVS is the same as a vector space internal to TopTopprovided that we use the two-sorted notion of vector space (k,X)(k, X) so that the first sort is interpreted as the topological ground field. (More generally, there is a notion of topological module which is the internalization in TopTop of the two-sorted notion of module involving a ring sort and a module over that ring; this notion is purely equational and thus interpretable in terms of finite products. Then a TVS is a topological module whose underlying ring is a field.) N.B.: if we internalize the one-sorted notion vector space, in which the ground field is encoded in the Lawvere theory as the set of unary operations, the continuity of the unary operations gives only a continuous map |k|×XX{|k|} \times X \to X where we use the discrete topology on kk – which is not what we want.

Like any topological abelian group, a TVS XX carries a uniform space structure generated by a basis of entourages (aka vicinities) that correspond to neighborhoods UU of 00:

{(u,v)X×X:uvU}\{(u, v) \in X \times X: u - v \in U\}

Thus many uniform notions (uniform continuity, completeness, etc.) carry over to the TVS context. Also from the uniformity (although it is also easy to prove directly), it follows that a TVS is completely regular, and also Hausdorff if and only if it is T 0T_0 (see separation axiom). Most authors insist on the T 0T_0 condition to rule out degenerate cases, but that prevents the category of TVSes from being topological over Vect. If the TVS VV is not Hausdorff, then the subset V 0V_0 defined as the intersection of all neighborhoods of zero is a vector subspace of VV and the quotient vector space V/V 0V/V_0 is Hausdorff, hence Tihonov (= completely regular Hausdorff).

The condition that scalar multiplication is continuous puts significant constraints on the topology of XX. For example, local compactness of kk implies, when VV is Hausdorff, that for any non-zero vXv \in X the function

v:kV- \cdot v: k \to V

maps kk homeomorphically onto its image. It follows quickly that XX cannot (for instance) be compact (unless it is the zero space and so has no non-zero vv); a classical theorem along these lines is that VV (over a local field) can be locally compact Hausdorff if and only if VV is finite-dimensional. (In the non-Hausdorff case, the theorems are that XX is compact if and only if its topology is indiscrete and that XX is locally compact if and only if it is a finitary direct sum of indiscrete spaces.) On the other hand, a nice property of even infinite-dimensional TVSes is that they are path-connected, or at least so in the classical cases where the ground field is \mathbb{R} or \mathbb{C}.

More classical material should be added, particularly on locally convex spaces.

TVSes from a Hilbert space viewpoint

The theory of TVS can be understood as the quest to find the essence of many fundamental theorems of functional analysis of Hilbert spaces (or Banach spaces), namely to find the minimal set of assumptions that are needed for Hilbert space theorems to remain true. Examples of these are:

A central rôle in the whole theory plays duality, that is the study of locally convex spaces and their duals. A prominent example is the definition of certain concepts by duality in the theory of Schwartz distribution?s.

Important subclasses

Topological vector spaces come in many flavours. The following chart provides a first overview (chart originally created and published by Greg Kuperberg on MathOverflow here, current version generated using Graphviz from lctvs dot source):

LCTVS FD Finite-Dimensional Hi Hilbert FD->Hi SC Second-Countable FD->SC Nu Nuclear FD->Nu Mo Montel FD->Mo Ba Banach Hi->Ba IP Inner-Product Hi->IP Re Reflexive Hi->Re Se Separable SC->Se Me Metrisable SC->Me Sc Schwartz Nu->Sc UB Ultrabornological Ba->UB Fr Fréchet Ba->Fr DF DF Ba->DF No Normed Ba->No IP->No Mo->Re Pc Paracompact Mo->Pc Bo Bornological UB->Bo Cn Convenient Fr->Cn Cp Complete Fr->Cp Br Baire Fr->Br Fr->Me SR Semi-Reflexive Re->SR Bl Barrelled Re->Bl Cn->Bo LC Locally Complete Cn->LC Cp->LC QC Quasi-Complete Cp->QC Br->Bl Me->Bo Me->Pc SR->QC QB Quasi-Barralled Bo->QB Bl->QB Sp Sequentially Complete QC->Sp Nm Normal Pc->Nm Mk Mackey QB->Mk CP Countably Paracompact Nm->CP

References

Wikipedia already has many nice references.

category: analysis

Revised on July 9, 2014 19:35:58 by Todd Trimble (98.208.182.196)