Editing Topological vector space (section)

==Types==

Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the [[closed graph theorem]], the [[Open mapping theorem (functional analysis)|open mapping theorem]], and the fact that the dual space of the space separates points in the space.

Below are some common topological vector spaces, roughly in order of increasing "niceness."

* [[F-space]]s are [[complete space|complete]] topological vector spaces with a translation-invariant metric.{{sfn|Rudin|1991|p=9 §1.8}} These include [[Lp space|<math>L^p</math> spaces]] for all <math>p > 0.</math>
* [[Locally convex topological vector space]]s: here each point has a [[local base]] consisting of [[convex set]]s.{{sfn|Rudin|1991|p=9 §1.8}} By a technique known as [[Minkowski functional]]s it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms.{{sfn|Rudin|1991|p=27 Theorem 1.36}} Local convexity is the minimum requirement for "geometrical" arguments like the [[Hahn–Banach theorem]]. The <math>L^p</math> spaces are locally convex (in fact, Banach spaces) for all <math>p \geq 1,</math> but not for <math>0 < p < 1.</math>
* [[Barrelled space]]s: locally convex spaces where the [[Banach–Steinhaus theorem]] holds.
* [[Bornological space]]: a locally convex space where the [[continuous linear operator]]s to any locally convex space are exactly the [[bounded linear operator]]s.
* [[Stereotype space]]: a locally convex space satisfying a variant of [[reflexive space|reflexivity condition]], where the dual space is endowed with the topology of uniform convergence on [[totally bounded space|totally bounded sets]].
* [[Montel space]]: a barrelled space where every [[closed set|closed]] and [[Bounded set (topological vector space)|bounded set]] is [[compact set|compact]]
* [[Fréchet space]]s: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class -- <math>C^\infty(\R)</math> is a Fréchet space under the seminorms <math display=inline>\|f\|_{k,\ell} = \sup_{x\in[-k,k]} |f^{(\ell)}(x)|.</math> A locally convex F-space is a Fréchet space.{{sfn|Rudin|1991|p=9 §1.8}}
* [[LF-space]]s are [[limit (category theory)|limits]] of [[Fréchet space]]s. [[ILH space]]s are [[inverse limit]]s of Hilbert spaces.
* [[Nuclear space]]s: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a [[nuclear operator]].
* [[Normed space]]s and [[seminormed space]]s: locally convex spaces where the topology can be described by a single [[norm (mathematics)|norm]] or [[seminorm (mathematics)|seminorm]]. In normed spaces a linear operator is continuous if and only if it is bounded.
* [[Banach space]]s: Complete [[normed vector space]]s. Most of functional analysis is formulated for Banach spaces. This class includes the <math>L^p</math> spaces with <math>1\leq p \leq \infty,</math> the space <math>BV</math> of [[Bounded variation|functions of bounded variation]], and [[Ba space|certain spaces]] of measures.
* [[Reflexive space|Reflexive Banach space]]s: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is {{em|not}} reflexive is [[Lp space|<math>L^1</math>]], whose dual is <math>L^{\infty}</math> but is strictly contained in the dual of <math>L^{\infty}.</math>
* [[Hilbert space]]s: these have an [[inner product]]; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include <math>L^2</math> spaces, the <math>L^2</math> [[Sobolev space|Sobolev spaces]] <math>W^{2,k},</math> and [[Hardy space|Hardy spaces]].
* [[Euclidean space]]s: <math>\R^n</math> or <math>\Complex^n</math> with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite <math>n,</math> there is only one <math>n</math>-dimensional topological vector space, up to isomorphism. It follows from this that any finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).