nLab
topological vector space

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\}

What the heck is entourage ? I found the word incomprehensible, hence useless. P.S. Aha founded the terminology under uniform space. Maybe one should create an entry for that (and include alternative words to the French one). Russian word okruzhenie (I learend uniform spaces only from the Russian edition of Engelking) would be in English something like encircling (etimologically) or environment (in other usages). I am not familiar with the standard in English. For the basis you quote Engelking seems to say “a basis for the unformity” –Zoran

Toby: The usual term in English is ‘entourage’; another term used is ‘vicinity’. Since ‘entourage’ is a perfectly good English word in ordinary usage, the translators of Bourbaki saw no need to change it from the French, and that is how it came into English mathematical usage. One reason for writing ‘basis of entourages’ instead of ‘basis for the uniformity’ is to clarify that these are entourages rather than uniform covers; however, since one usually approaches uniform spaces through entourages instead of through uniform covers, I think that ‘basis for the uniformity’ would be fine here, if you want to change it. (I have added ‘vicinity’ to uniform space; someday, I might even add uniform covers there too!)

Zoran: when I read above passage I did not even know that entourage is a technical term (I first thought this is a poetic expression) hence only after a while starting looking into uniform space. Another newcomer might have the same confusion unless oen creates and links to entourage?.

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.