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locally convex topological vector space

Contents

Definition

A topological vector space is locally convex if it has a base of its topology consisting of convex open subsets. Equivalently, it is a vector space equipped with a gauge consisting of seminorms. As with other topological vector spaces, a locally convex space (LCS or LCTVS) is often assumed to be Hausdorff.

Locally convex (topological vector) spaces are the standard setup for much of contemporary functional analysis.

A natural notion of smooth map between lctvs is given by Michal-Bastiani smooth maps.

Properties

Continuous linear functionals

One reason why locally convex topological vector spaces are important is that lots of continuous linear functionals exist on them, at least if one assumes an appropriate choice principle, e.g., axiom of choice or ultrafilter theorem (or just dependent choice for a separable space). This fact is encapsulated in the Hahn-Banach theorem; a nice exposition is given in Tao 09. By way of contrast, a topological vector space which is not locally convex, such as the Lebesgue space L p([0,1])L^p([0, 1]) where 0<p<10 \lt p \lt 1, need not have any (nonzero) continuous linear functionals at all.

Definition

(directed system of seminorms)

A family {p q} qQ\{p_q\}_{q \in Q} of seminorms on some real vector space VV is called directed if

q 1,q 2Q(qQ(C(0,)(vV(Cp q(v)max{p q 1(v),p q 2(v)})))). \underset{q_1, q_2 \in Q}{\forall} \left( \underset{q \in Q}{\exists} \left( \underset{C \in (0,\infty)}{\exists} \left( \underset{v \in V}{\forall} \left( \, C p_q(v) \leq max\{ p_{q_1}(v), p_{q_2}(v) \} \, \right) \right) \right) \right) \,.

(e.g. Infusino 17 def. 4.2.15)

Proposition

(maximum envelope of seminorms)

Let VV be a real vector space equipped with a set {p q:V} qQ\{p_q \colon V \to \mathbb{R}\}_{q \in Q} of seminorms.

Then the maxima of non-empty finite subsets JQJ \subset Q of these seminorms

v(maxqJp q(v)) v \mapsto \left( \underset{q \in J}{max} p_q(v) \right)

are themselves seminorms, and the set of them

{maxqJp q()} JQfinite, non-empty \left\{ \underset{q \subset J}{max} p_q(-) \right\}_{ { J \subset Q } \atop {\text{finite, non-empty}} }

generate the same topology on VV as the original {p q}\{p_q\} do. Moreover, the system of maximum seminorms evidently form a filtered system according to def. 1.

(e.g. Infusino 17 prop. 4.2.13, 4.2.14)

Proposition

(characterization of continuity for linear functionals by norm-bounds)

Let VV be a real vector space and τ\tau a topology on VV that makes it a locally convex topological vector space, induced from a set of seminorms {p q:V} qQ\{p_q \colon V \to \mathbb{R}\}_{q \in Q}. Consider a linear function

L:V L \colon V \to \mathbb{R}
  1. (directed system of seminorms) If the system of seminorms {p q} qQ\{p_q\}_{q \in Q} is directed (def. 1) then LL is a continuous function with respect to τ\tau, hence is a continuous linear functional, precisely if it is qq-continuous for some qQq \in Q:

    (Lcontinuous with respect toτ)qQ(C(0,)(vV(|L(v)|Cp q(v)))). \left( L \,\text{continuous with respect to}\, \tau \right) \;\Leftrightarrow\; \underset{q \in Q}{\exists} \left( \underset{C \in (0,\infty)}{\exists} \left( \underset{v \in V}{\forall} \left( \, {\vert L(v)\vert} \leq C p_q(v) \, \right) \right) \right) \,.
  2. (general system of seminorms) Together with prop. 1 this means that for {p q} qQ\{p_q\}_{q \in Q} any set of seminorms (not necessarily directed), then LL is continuous precisely if there exists a inhabited finite subset of seminorms such that LL is bounded with respect to the maximum over this set:

    (Lcontinuous with respect toτ)q 1,,q nQ(C(0,)(vV(|L(v)|Cmaxk=1,,np q n(v)))). \left( L \,\text{continuous with respect to}\, \tau \right) \;\Leftrightarrow\; \underset{q_1, \cdots, q_n \in Q}{\exists} \left( \underset{C \in (0,\infty)}{\exists} \left( \underset{v \in V}{\forall} \left( \, {\vert L(v)\vert} \leq C \underset{k = 1, \cdots, n}{max} p_{q_n}(v) \, \right) \right) \right) \,.

(e.g. Infusino 17, prop. 4.6.1, corollary 4.6.2), or remark 3-4 4. here: pdf

Co-Probes and curves

The collections of continuous linear functionals on a LCTVS is used in a way analogous to the collection of coordinate projections pr i: npr_i:\mathbb{R}^n\to \mathbb{R} out of a Cartesian space. For example, curves in a LCTVS over the reals can be composed with functionals to arrive at a collection of functions \mathbb{R} \to \mathbb{R} which are analogous to the ‘components’ of the curve.

In one respect, a locally convex TVS is a nice topological space in that there are enough co-probes by maps to the base field.

Diagram of properties

LCTVS cluster_key_col1 cluster_key_col2 cluster_key_col3 FD FD Hi Hi FD->Hi NuFr FD->NuFr IP IP Hi->IP ReBa Hi->ReBa Nu Nu Sc Sc Nu->Sc Ba Ba Fr Fr Ba->Fr No No Ba->No LB LB Ba->LB IP->No Mo Mo Re Re Mo->Re UB UB Bo Bo UB->Bo QCQB UB->QCQB LF LF Fr->LF Me Me Fr->Me Pt Pt Fr->Pt DF DF No->DF No->Me QB QB Bo->QB LF->UB LB->DF LB->LF Me->Bo NuFr->Nu NuFr->Mo ReFr NuFr->ReFr LC LC QC QC Sq Sq QC->Sq BC BC Pt->BC Cp Cp BC->Cp Sq->LC Cp->QC Bl Bl Re->Bl MkSR Re->MkSR SR SR SR->QC Mk Mk QB->Mk Bl->QB MkSR->SR MkSR->Mk QCQB->QC QCQB->Bl ReBa->Ba ReBa->ReFr ReFr->Fr ReFr->Re yFD yHi xFD FD: Finite-Dimensional yDF Key to symbols yNo xDF DF: DF yPt yBC xPt Pt: Ptak Space yNu xHi Hi: Hilbert (technically, admits a Hilbertian structure) yBa xNu Nu: Nuclear yIP xBa Ba: Banach (technically, complete and normable) yMo xIP IP: Topology from an inner-product ySc xMo Mo: Montel yUB xSc Sc: Schwartz yFr xUB UB: Ultrabornological yZ1 xFr Fr: Fréchet yBo xNo No: Normable space yLF xBo Bo: Bornological yLB xLF LF: strict inductive sequence of Fréchet spaces yMe xLB LB: strict inductive sequence of Banach spaces yLC xMe Me: Metrisable yQC xLC LC: Locally Complete yZ2 xQC QC: Quasi-Complete ySq xBC BC: Br Space yCp xSq Sq: Sequentially Complete yRe xCp Cp: Complete ySR xRe Re: Reflexive yQB xSR SR: Semi-Reflexive yMk xQB QB: Quasi-Barrelled yBl xMk Mk: Mackey yZ3 xBl Bl: Barrelled

Examples

References

category: analysis

Revised on October 23, 2017 08:58:14 by Urs Schreiber (88.78.246.194)