nLab
locally convex space

Contents

Definition

A topological vector space is locally convex if it has a base of 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) is often assumed to be Hausdorff space.

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

Properties

Functionals

One reason why locally convex TVS 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 Terry Tao’s lecture notes. By way of contrast, a TVS which is not locally convex, such as the topological vector space L p([0,1])L^p([0, 1]) where 0<p<10 \lt p \lt 1, need not have any (nonzero) functionals at all.

The collections of 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}. 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 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

References

  • J. L. Taylor, Notes on locally convex topological vector spaces (1995) (pdf)

category: analysis

Revised on October 12, 2013 21:27:08 by Toby Bartels (98.19.41.253)