Basic Analysis – Gently Done Topological Vector Spaces

70 Basic Analysis
vector topology on R is determined by this neighbourhood base at 0 together with
all its translates. This is, in fact, strictly finer than the σ(R,{p})-topology.
This observation is valid in general. If P is any family of seminorms on a vector
space X (�= {0}), then the σ(X,P)-topology has subbase given by the sets of the
form {x : p(x) ∈ (s,t)} for p ∈ P and s < t. Each of these sets is symmetrical
about0inX, andsothisproperty persistsforallnon-empty setsopenwithrespect
to the σ(X,P)-topology. This topology is never Hausdorff—the points x and −x
cannot be separated, for any x �= 0. However, if p ∈ P and if p(x) = r �= 0, then
V(x,p;r) and V(−x,p;r) are disjoint open neighbourhoods of the points x and
−x, respectively, with respect to the vector topology determined by P. (Of course,
if P is not separating, it will not be possible to separate x and −x for all points
x in X. In fact, if p(x) = 0 for some x �= 0 in X and all p ∈ P, we see that every
V(x,p;r) contains −x.)
We can rephrase this in terms of the continuity of the seminorms. We know
that a net x ν converges to x in X with respect to the σ(X,P)-topology if and
only if p(x ν ) → p(x) for each p ∈ P. As we have noted earlier, the convergence of
p(x ν ), for every p ∈ P, does not imply the convergence of x ν with respect to the
vector topology determined by the family P.
The sequence p((−1) n x) converges to p(x) = p(−x) for any p ∈ P and any
x ∈ X, and so itfollowsthat(−1) n x converges bothto x andto −x withrespect to
the σ(X,P)-topology. This cannot happen in the vector topology for any pair x,p
with p(x) �= 0—because, as noted above, in this case x and −x can be separated.
Theorem 7.8 Suppose that ρ is a seminorm on a topological vector space (X,T).
The following statements are equivalent.
(i) ρ is continuous at 0.
(ii) ρ is continuous.
(iii) ρ is bounded on some neighbourhood of 0.
If the topology T on X is the vector topology determined by a family P of seminorms,
then (i), (ii) and (iii) are each equivalent to the following.
(iv) There is a finite set of seminorms p 1 ,...,p m in P and a constant C > 0
such that
ρ(x) ≤ C(p 1 (x)+···+p m (x)) for x ∈ X.
Proof The inequality |ρ(x)−ρ(y)| ≤ ρ(x−y) implies that (ii) follows from (i)—if
x ν → x in X, then x ν −x → 0 so that |ρ(x ν )−ρ(x)| ≤ ρ(x ν −x) → 0, i.e., ρ is
continuous at x.
Clearly (ii) implies (i), and (iv) implies (i)—if x ν is any net converging to 0,
then p(x ν ) → 0 for each p in P, so that ρ(x ν ) → 0 = ρ(0) by (iv).
Department of Mathematics King’s College, London