480 CHAPTER 18. TOPOLOGICAL VECTOR SPACES

Theorem 18.1.3 The vector space operations of addition and scalar multiplication arecontinuous. More precisely,

+ : X×X → X , · : F×X→X

are continuous.

Proof: It suffices to show +−1(B) is open in X ×X and ·−1(B) is open in F×X if B isof the form

B = {y ∈ X : ρ(y− x)< r}

because finite intersections of such sets form the basis B. (This collection of sets is asubbasis.) Suppose u+ v ∈ B where B is described above. Then

ρ(u+ v− x)< λ r

for some λ < 1. ConsiderBρ(u,δ )×Bρ(v,δ ).

If (u1,v1) is in this set, then

ρ(u1 + v1− x) ≤ ρ (u+ v− x)+ρ (u1−u)+ρ(v1− v)

< λ r+2δ .

Let δ be positive but small enough that

2δ +λ r < r.

Thus this choice of δ shows that +−1(B) is open and this shows + is continuous.Now suppose αz ∈ B. Then

ρ(αz− x)< λ r < r

for some λ ∈ (0,1). Let δ > 0 be small enough that δ < 1 and also

λ r+δ (ρ (z)+1)+δ |α|< r.

Then consider (β ,w) ∈ B(α,δ )×Bρ (z,δ ).

ρ (βw− x)−ρ (αz− x) ≤ ρ (βw−αz)

≤ |β −α|ρ (w)+ρ (w− z) |α|≤ |β −α|(ρ (z)+1)+ρ (w− z) |α|< δ (ρ (z)+1)+δ |α|.

Henceρ (βw− x)< λ r+δ (ρ (z)+1)+δ |α|< r

and soB(α,δ )×Bρ (z,δ )⊆ ·−1(B).

This proves the theorem.