Basic Analysis – Gently Done Topological Vector Spaces

68 Basic Analysis
To show that each p ∈ P is continuous, let ε > 0 and suppose that x ν → x in
(X,T). Then there is ν 0 such that x ν ∈ V(x,p;ε) whenever ν � ν 0 . Hence
|p(x)−p(x ν )| ≤ p(x−x ν ) < ε
whenever ν � ν 0 and it follows that p : X → R is continuous.
It remains to show that (X,T) is Hausdorff if and only if P is a separating
family. Suppose that P is separating, and let x,y ∈ X with x �= y. Then there
) are
is some p ∈ P such that δ = p(x−y) > 0. The sets V(x,p; δ
2
) and V(y,p; δ
2
disjoint neighbourhoods of x and y so that (X,T) is Hausdorff.
Conversely, suppose that (X,T) is Hausdorff (which follows, as we have seen,
from the assumption that each one-point set be closed). For any given x ∈ X,
with x �= 0, there is a neighbourhood of 0 not containing x. In particular, there
is p 1 ,...,p m ∈ P and r > 0 such that x /∈ V(0,p 1 ,...,p m ;r). It follows that
p i (x−0) = p i (x) ≥ r, for some 1 ≤ i ≤ m, and so certainly p i (x) > 0 and we see
that P is a separating family of seminorms on X.
Definition 7.2 The topology T on a vector space X over K constructed above is
called the (vector space) topology determined by the given family of seminorms P.
Theorem 7.3 Let T be the vector space topology on a vector space X determined
by a family P of seminorms. A net (x ν ) converges to 0 in (X,T) if and only if
p(x ν ) → 0 for each p ∈ P.
Proof Suppose that x ν → 0 in (X,T). Then p(x ν ) → p(0) = 0 for each p ∈ P,
since each such p is continuous.
Conversely, suppose that p(x ν ) → 0 for each p ∈ P. Let p 1 ,...,p m ∈ P and let
r > 0. Then there is ν 0 such that p i (x ν ) < r whenever ν � ν 0 , 1 ≤ i ≤ m. Hence
x ν ∈ V(0,p 1 ,...,p m ;r) whenever ν � ν 0 . It follows that x ν → 0.
Remark 7.4 The convergence of a net (x ν ) to x is not necessarily implied by the
convergence of p(x ν ) → p(x) in R for each p ∈ P. Indeed, for any x �= 0 and any
p ∈ P, p((−1) n x) → p(x), as n → ∞, but it is not true that (−1) n x → x if (X,T)
is separated. We will come back to this observation later.
Department of Mathematics King’s College, London