Basic Analysis – Gently Done Topological Vector Spaces

6. Topological Vector Spaces
There are many situations in which we encounter a natural linear structure as well
as a topological one; R n and C([0,1]) being obvious examples. The harmonious
coalescence of linearity and topological constructs is realized in the concept of
topological vector space.
Definition 6.1 A topological vector space over K is a vector space X over K
furnished with a topology T such that
(i) the map (x,y) ↦→ x+y is continuous from X ×X into X (where X ×X is
given the product topology);
(ii) the map (t,x) ↦→ tx is continuous from K × X into X (where K has its
usual topology, and K×X the product topology).
One says that T is a vector topology on the vector space X, or that T is compatible
with the linear structure of X.
We say that a topological vector space (X,T) is separated if the topology T is a
Hausdorff topology.
Inwords, atopologicalvectorspaceisavectorspacewhichisatthesametimea
topological space such that addition and scalar multiplication are continuous. We
will see later that, as a consequence of the compatibility between the topological
and linear structure, a topological vector space is Hausdorff if and only every onepoint
set is closed. Sometimes the requirement that the topology be Hausdorff is
taken as part of the definition of a topological vector space.
Example 6.2 Any real or complex normed space isatopologicalvector space when
equipped with the topology induced by the norm.
Any vector space is a topological vector space when equipped with the indiscrete
topology. Of course, this will fail to be a separated topological vector space unless
it is the zero-dimensional space {0}.
It is convenient to introduce some notation at this point. Basically, we simply
wish to extend the vector space notation for addition and scalar multiplication to
the obvious thing for subsets. Let X be a vector space over K and let A and B be
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