Basic Analysis – Gently Done Topological Vector Spaces

7: Locally Convex Topological Vector Spaces 83
2. Denote by (s) the linear space of all complex sequences (x n ) equipped with the
family P of seminorms P = {p k : k ∈ N}, where p k ((x n )) = |x k |. Convergence in
T P is componentwise convergence.
For each k, let ℓ k : (s) → C be the map ℓ k ((x n )) = x k . Then ℓ k is a linear
functional and T P is equal to the σ((s),F)-topology, where F = {ℓ k : k ∈ N}.
3. We can extend the previous example somewhat. Let X be the linear space
of maps f from a given set Ω into C. For each ω ∈ Ω, let p ω (f) = f(ω) and let
P = {p ω : ω ∈ Ω}. The topology T P on X is the topology of pointwise convergence
on Ω. If Ω = N, then this example reduces to the previous one.
4. The space S(R) is the space of infinitely differentiable complex-valued functions
f on R such that for each m,n = 0,1,2,..., the set {|x| m |D n f(x)| : x ∈ R}
is bounded. For such f, set p m,n (f) = sup(1 + |x| m )|D n f(x)| : x ∈ R}. Then
P = {p m,n : m,n ∈ N} is a separating family of seminorms. S(R) is the Schwartz
space of smooth functions of rapid decrease. In a similar way, one defines S(R n ),
the space of smooth functions on R n with rapid decrease. These spaces play an
important rôle in the general theory of partial differential equations and also in
quantum field theory.
5. Let X = B(H), the linear space of bounded operators on a Hilbert space H.
For each x ∈ H, let p x be the seminorm p x (T) = �Tx�, for T ∈ B(H), and set
P = {p x : x ∈ H}. Then P is a separating family and T P is the topology of strong
operator convergence, T ν → T if and only if �T ν x−Tx� → 0, for each x ∈ H.
6. Let X = B(H), as above. For each x,y ∈ H, let ℓ x,y be the linear functional
on X given by ℓ x,y (T) = 〈Tx,y〉 and let p x,y be the seminorm p x,y (T) = |ℓ x,y (T)|.
Let P be the collection of all such p x,y with x,y ∈ H and let F be the collection of
the linear functionals ℓ x,y . The topology T P is equal to the σ(B(H),F)-topology—
it is the weak operator topology. A net T ν converges to T with respect to this
topology if and only if 〈T ν x,y〉 → 〈Tx,y〉, for each pair x,y ∈ H.