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Locally Convex Vector Spaces III: the Metric Point of View LCVS III c Gabriel Nagy Locally Convex Vector Spaces III: The Metric Point of View Notes from the Functional Analysis Course (Fall 07 - Spring 08) Warning! Some proofs are based on Exercises from previous lectures. The reader is urged to solve all Exercises from CW I-III and LCVS I-II. (See Remarks 1-6 below.) Convention. Throughout this note K will be one of the fields R or C, and all vector spaces are over K. A. Locally convex topologies defined by seminorms In this sub-section we outline a method of constructing locally convex topologies using seminorms. Notation. Given a vector space X and a seminorm p on X , we define the two unit p-balls as the sets: BX (p) = {x ∈ X : p(x) < 1}, BX (p) = {x ∈ X : p(x) ≤ 1}. (When there is no danger of confusion, the subscript X will be omitted from the notation.) Remarks 1-7. Most of the properties listed below are essentially contained in CW II (see the brief references provided) 1. The sets B(p) and B(p). are convex and balanced. The set B(p) is openly absorbing, and its Minkowski functional is: qB(p) = p. (Exercise 8 from CW II.) 2. If A ⊂ X is convex and openly absorbing, then A can be recovered from its Minkowski functional as: A = {x ∈ X : qA(x) < 1}. In particular, if A is also balanced, then qA is a seminorm, and one has the equality A = B(qA). (Proposition 2 from CW II). 3. A sufficient condition for a subset 0 ∈ A ⊂ X to be openly absorbing is that A is open with respect to some linear topology on X . (Exercise 6 from CW II.) 4. For two seminorms p and q, the following are equivalent: (i) p(x) ≤ q(x), ∀ x ∈ X ; (ii) B(p) ⊃ B(q). 1 The implication (i) ⇒ (ii) is trivial. For the implication (ii) ⇒ (i), observe that for 1 q(x) every x ∈ X and every ε > 0, one has q q(x)+ε x = q(x)+ε < 1, so if (ii) holds then this 1 implies also p q(x)+ε x ≤ 1, i.e. p(x) ≤ q(x) + ε, ∀ ε > 0, hence (i) follows. 5. If p is a seminorm on X then εB(p) = {x ∈ X : p(x) < ε}, εB(p) = {x ∈ X : p(x) ≤ ε}. Clearly one has x ∈ εB(p) ⇔ ε−1x ∈ B(p) ⇔ p(ε−1x) < 1 ⇔ p(x) < ε, and similar equivalences with ≤ in place of <. 6. If p is a seminorm on X then |p(x) − p(y)| ≤ p(x − y), ∀ x, y ∈ X . Use p(x − y) = p(y − x), and the triangle inequalities p(y) + p(x − y) ≥ p(x) and p(x) + p(y − x) ≥ p(y). 7. If X is equipped with a linear topology T, and p is a seminorm on X , then the following are equivalent: (i) p is continuous; (ii) p is continuous at 0; (iii) B(p) is a neighborhood of 0; (iii’) B(p) is a neighborhood of 0. First of all, the implications (i) ⇒ (iii) ⇒ (iii0) are trivial. For the implication (iii0) ⇒ (ii), we must show that, assuming (iii’), for every ε > 0, the set V = {x ∈ ε X : p(x) < ε} is a neighborhood of 0. But this is trivial, since V ⊃ 2 B(p). Finally, for the implication (ii) ⇒ (i), we notice that if p is continuous at 0, and xλ → x, then (xλ − x) → 0, and then the inequalities 0 ≤ |p(xλ) − p(x)| ≤ p(xλ − x) will clearly force p(xλ) → p(x). The starting observation in this section is contained in the following result. Proposition 1. Let X be a locally convex vector space. (i) For every balanced convex neighborhood A of 0, the Minkowski functional qA is a con- tinuous seminorm on X . (ii) The correspondence A open, convex, q continuous A ⊂ X 3 A 7−→ qA ∈ q : X → [0, ∞) (1) balanced, A 3 0 seminorm on X is bijective. Its inverse is the map: 2 q continuous A open, convex, q : X → [0, ∞) 3 p 7−→ B(p) ∈ A ⊂ X . (2) seminorm on X balanced, A 3 0 Proof. (i). Fix some balanced convex neighborhood A of 0. We already know (CW II) that qA is a seminorm on A, and A = B(qA). By Remark 7, qA is continuous. (ii). For simplicity, let B denote the source set of (1), and let P denote the collection of all continuous seminorms. First of all, for every p ∈ P, the unit ball B(p) is convex and openly absorbing (by Remark 1). Furthermore, B(p) is also open, by continuity, so the correspondence (2) indeed takes values in B. Secondly, by Remarks 1 and 2, we also know that qB(p) = p, ∀ p ∈ P; B(qA) = A, ∀ A ∈ C, which clearly show that (1) and (2) are inverses of each other. Theorem-Definition 1. Let X be a vector space, and let P be a non-empty collection of seminorms on X . If we define, for every ε > 0 and p ∈ P the set Up.ε = {x ∈ X : p(x) < ε}, there exists a unique locally convex topology T on X , such that the collection U = {Up.ε : p ∈ P, ε > 0} constitutes a fundamental system of T-neighborhoods of 0. Moreover: (i) all p ∈ P are T-continuous; (ii) T is the weakest among all locally convex topologies on X which make all p ∈ P are continuous. The topology T is referred to as the locally convex topology defined by P, as is denoted by T(P). If P is a singleton {p}, we denote this topology simply by T(p). Proof. Consider the collection C = {B(p): p ∈ P}. Since all sets in C are convex, balanced, and (openly) absorbing, by Theorem 1 from LCVS I, there exists a unique locally convex topology T, for which the collection1 U = {εB(p): ε > 0, p ∈ P} (3) is a fundamental system of T-neighborhoods of 0. To prove (i) we simply notice that, since every p ∈ P can be written (by Remark 1) as the Minkowski functional of its unit ball p = qB(p), and B(p) is by construction a convex balanced T-neighborhood of 0, the contituity of p follows from Proposition 1(i). To prove (ii), fix another locally convex topology T0 such that all p ∈ P are T0-continuous, and let us show that T0 ⊃ T. Since both T and T0 are linear, all we need to show is that: 1 By Remark 5, εB(p) = Up,ε. 3 (∗) every T-neighborhood of 0 is also a T0-neighborhood of 0. Since T has U as a fundamental system of T-neighborhoods for 0, the above condition is equivalent to the condition that every set in U is a T0-neighborhood of 0. Using (3) and the fact that dilations are homeomorphisms in linear topologies we now see that (∗) is in fact equivalent to the following. (∗∗) For every p ∈ P, the unit ball B(p) is a T0-neighborhood of 0. But (∗∗) is clearly true, since all p ∈ P are T0-continuous (so the B(p)’s are in fact open neighborhoods of 0.) Remark 8. With note notations as above, if we consider, for every p ∈ P, ε > 0, the set Wpε = {x ∈ X : p(x) ≤ ε} = εB(p), then the collection W = {Wpε : p ∈ P, ε > 0} also constitutes a fundamental system of T-neighborhoods of 0. This is pretty obvious from the inclusions Upε ⊂ Wpε ⊂ Up,2ε. Remark 9. Every locally convex topology on a vector space X can be constructed as T(P), for a suitably chosen collection P of seminorms on X . More precisely, if S is a locally convex topology, we can simply take P to be the collection of all S-continuous seminorms. On the one hand, by Theorem-Definition 1, since all p ∈ P are S-continuous, we automatically have the inclusion S ⊃ T(P). On the other hand, (∗) every S-neighborhood of 0 is also a T(P)-neighborhood of 0, so we also have the inclusion S ⊂ T(P). Property (∗) can be proven as follows. Start with some S-neighborhood V of 0, and choose a balanced open convex set A ⊂ V. By Proposition 1(i), the Minkowski functional qA is S-continuous, hence qA belongs to P. By Remark 2, A = B(qA), so by the definition of T(P), it follows that A is a T(P)-neighborhood of 0, and so will be V ⊃ A. Remark 10. If a locally convex space X has its topology defined by a family P of semi- norms, the for a net (xλ)λ∈Λ in X and some x ∈ X , the following conditions are equivalent: (i) xλ → x; (ii) p(xλ − x) → 0, ∀ p ∈ P. Of course, the implication (i) ⇒ (ii) is trivial. Conversely, if condition (ii) holds, then for every p ∈ P and every ε > 0, there exists λp,ε ∈ Λ, such that p(xλ − x) < ε, ∀ λ λp,ε. Equivalently, using the notations from Theorem-Definition 1, it follows that for every U ∈ U, there exists λU such that xλ ∈ U + x, ∀, λ λU , and since by construction the collection {U + x : U ∈ U} is a fundamental system of neighborhoods for x, it follows that (xλ) indeed converges to x. 4 Remark 11. Given a family P of seminorms on X , the locally convex topology T(P) is Hausdorff, if and only if the “null set” N (P) = {x ∈ X : p(x) = 0, ∀ p ∈ P} is equal to the singleton set {0}.
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