nLab
topological vector space

Idea

A topological vector space, or TVS for short, is a vector space $X$ (usually over the ground field $k=ℝ$ or $k=ℂ$) equipped with a topology for which the addition and scalar multiplication maps

$$+:X\times X\to X,\cdot :k\times X\to X$$+: X \times X \to X, \qquad \cdot: k \times X \to X

are continuous (where $k$ is given its standard topology).

Much as a topological group is a group object in Top, so a TVS is a vector space internal to $\mathrm{Top}$ … but not just any vector space in $\mathrm{Top}$ is a TVS! The reason is that, in a vector space internal to $\mathrm{Top}$, $\cdot$ only need be continuous in the second variable; in other words, this concept uses the discrete topology on $k$. So only some vector spaces in $\mathrm{Top}$ are TVSes.

Like any topological abelian group, a TVS $X$ carries a uniform space structure generated by a basis of entourages (aka vicinities) that correspond to neighborhoods $U$ of $0$:

$$\{(u,v)\in X\times X:u-v\in U\}$$\{(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 $T_0$ (see separation axiom). Most authors insist on the $T_0$ condition to rule out degenerate cases, but that prevents the category of TVSes from being topological over Vect. If the TVS $V$ is not Hausdorff, then the subset $V_0$ defined as the intersection of all neighborhoods of zero is a vector subspace of $V$ and the quotient vector space $V/V_0$ is Hausdorff, hence Tihonov (= completely regular Hausdorff).

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

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

maps $k$ homeomorphically onto its image. It follows quickly that $X$ cannot (for instance) be compact (unless it is the zero space and so has no non-zero $v$); a classical theorem along these lines is that $V$ can be locally compact Hausdorff if and only if $V$ is finite-dimensional. (In the non-Hausdorff case, the theorems are that $X$ is compact if and only if its topology is indiscrete and that $X$ 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.

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:

• the Hahn–Banach Theorem

• the Open Mapping and closed graph theorems

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):

• locally convex spaces: where the Hahn-Banach theorem works (assuming sufficient axioms)

• bornological topological vector spaces: where bounded means continuous

• barrelled topological vector spaces

• sequence space

References

Wikipedia already has many nice references.

category: analysis