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Home , Weak topology

Prof. Girardi 13. A Review of some Topology

13.1. Definition. Let X be a non-empty set. Let P(X) be the collection of all subsets of X, i.e. P(X) is the power set of X. Let τ ⊂ P(X). Then (X, τ) is a topological (or equivalently τ is a topology on X) provided • if A, B ∈ τ, then A ∩ B ∈ τ • for any index set Γ, if {Aα}α∈Γ ⊂ τ then ∪α∈ΓAα ∈ τ • ∅,X ∈ τ. In which case, A ∈ P(X) is an open set if and only if A ∈ τ. 13.2. Definition. Let (X, τ) be a and B ⊂ τ. Then B is a base (or basis) of τ provided ∀A ∈ τ ∃{Bα}α∈Γ ⊂ τ such that A = ∪α∈ΓBα . 13.3. Example. Let X be a normed linear space. As usual, for ε > 0 and x ∈ X, let

Nε(x) := {y ∈ X : kx − ykX < ε} . Let τ be the open subsets of X, i.e.,

τ = {A ∈ P(X): ∀x ∈ A, ∃ε > 0 such that Nε(x) ⊂ A} . Then a base B for (X, τ) is

B := {Nε(x): x ∈ X, ε > 0} . 13.4. Recall. Read through the course handouts, which are posted on the course homepage, for Section 13. 13.5. Definition. Let X be a over the field K where, as usual, K is R or C. • A linear functional on X is a linear mapping f : X → K. • Let Z be a family of linear functionals on X. Then Z separates points provided if x, y ∈ X and x 6= y then there is a f ∈ Z such that f(x) 6= f(y). • Let (X, τ) be a topological space. The topological dual of X, denoted by (X, τ)∗, is the set of all continuous linear functionals on X, i.e., ∗ (X, τ) := {f : X → K | f is continuous and linear} . 13.6. Definition/Recall. Let Z be a family of linear functionals that separates points on a vector space X. Then σ(X,Z) denotes the weakest (i.e. smallest) topology on X for which each f ∈ Z is continuous. • Clearly, each f ∈ Z is continuous with respect to the (large) topology σ(X, P(X)) on X. • If τ 0 is any topology on X such that each f ∈ Z is continuous with respect to (X, τ 0) then σ(X,Z) ⊂ τ 0. A base for σ(X,Z) is  n −1 ∩i=1fi (Ui): fi ∈ Z,Ui is open in K , n ∈ N . Another, more convenient, base for σ(X,Z) is  n −1 ∩i=1fi [Nε (fi (x0))] : fi ∈ Z , ε > 0 , x0 ∈ X , n ∈ N ; note that n −1 ∩i=1fi [Nε (fi (x0))] = {x ∈ X : |fi(x) − fi(x0)| < ε , i = 1, . . . , n}

= {x ∈ X : |fi(x − x0)| < ε , i = 1, . . . , n} =: Nε,f1,...,fn (x0) .

Since Z separates points, σ(X,Z) is a Hausdorff (i.e., T2) topology, i.e., if x1, x2 ∈ X and x1 6= x2 then there are disjoint U1,U2 ∈ σ(X,Z) such that xi ∈ Ui. Thus, if the limit of a from X exists, then the limit is unique. yr.mn.dy: 15.01.12 Page 1 of 4 Prof. Girardi 13. A Review of some Topology

13.7. Fact. Let X be a vector space. Let Z be a vector space of linear functionals on X that separates points. Then (X, σ (X,Z))∗ = Z. That (X, σ (X,Z))∗ ⊃ Z follows from the definition. And ⊂ follows from linear algebra (see [RS-I, Thm. IV.20]).

13.8. Optional Example. Let X be a over K, endowed with a k·kX and the norm topology τ as in Example 13.3. Let k·kX X∗ := X, τ ∗ = {x∗ : X → | x∗ is linear and (norm) continuous} . k·kX K (1) The σ(X,X∗) topology on X is called the . A basis for the weak topology consists of all sets of the form ∗ N ∗ ∗ (x) = {y ∈ X : |x (x − y)| < ε , for i = 1, . . . , n} . ε,x1,...,xn i ∗ ∗ ∗ where x ∈ X and x1, . . . , xn ∈ X and ε > 0 and n ∈ N. ∗ ∗ ∗ ε Note σ(X,X ) ⊂ τk·k since Nε,x ,...,xn (x) ⊃ N (x). X 1 ∗ max kxi k ∗ By Fact 13.7, (X, σ (X,X∗)) = X∗. So for a linear functional x∗ : X → K, x∗ weak continuous if and only if x∗ is norm continuous. (2) The σ(X∗,X) topology on X∗ is called the weak∗ topology. A basis for the weak∗ topology consists of all sets of the form ∗ ∗ ∗ ∗ ∗ Nε,x1...,xn (x ) = {y ∈ X : |(x − y )(xi)| < ε , for i = 1, . . . , n} . ∗ ∗ where x ∈ X and x1, . . . , xn ∈ X and ε > 0 and n ∈ N. Note σ(X∗,X) ⊂ σ(X∗,X∗∗) since X ⊂ X∗∗. ∗ By Fact 13.7, (X∗, σ (X∗,X)) = X. So for a linear functional x∗∗ : X∗ → K, x∗ weak∗ continuous if and only if x∗∗ comes from X, i.e, there is x ∈ X such that x∗∗(x∗) = x∗(x) for each x∗ ∈ X∗. 13.9. Optional Fact. Let X be a normed linear space. Then the following are equivalent. • dim X < ∞. • (X, σ (X,X∗)) is metrizable. • (X∗, σ (X∗,X)) is metrizable. Thus, often do not suffice and we need to consider nets. (See course handouts on nets.) 13.10. Definition. Let X be a vector space over K. Let ρ: X → [0, ∞) be a function. Then ρ is a (in German, Halbnorm) provided ∀x, y ∈ X and α ∈ K: • ρ(x + y) ≤ ρ(x) + ρ(y) • ρ(α x) = |α| ρ(x). Let’s think about this definition. Why might a seminorm not be a norm? Well, because ρ(x) = 0 need not imply that x = 0. So we cope with this badness by the following definition. A family {ρα}α∈A of separates points provided if ρα(x) = 0 for each α ∈ A then x = 0. 13.11. Definition. A space X is called a locally convex (i.e. (LCTVsp)) provided (1) X is a vector space over K (2) ∃ a family {ρα}α∈A of seminorms on X that separates points. In this case:

(3) the natural topology τ = τ{ρα}α∈A on X, generated by the seminorms {ρα}α∈A, is the weakest topology so that: • ρα is continuous for each α ∈ A • the mapping +: X × X → X, defined by (x, y) → x + y, is continuous . Note that since {ρα}α∈A separates points, τ is Hausdorff. yr.mn.dy: 15.01.12 Page 2 of 4 Prof. Girardi 13. A Review of some Topology

(4) a basis of τ is

{Nε,α1,...,αn (x) | αi ∈ A , ε > 0 , n ∈ N , x ∈ X}

where Nε,α1,...,αn (x) := {y ∈ X : ραi (x − y) < ε , i ∈ {1, . . . , n}}, which is a convex set (5) if {xβ}β∈B is a net in X, then

xβ → x ⇐⇒ ρα(xβ − x) → 0 ∀α ∈ A. 13.12. Example. Let X be a Banach space. The natural topology generated by the family of seminorms consisting of: • just one seminorm, namely the norm on X, is just the usual norm topology on X. ∗ ∗ •{ρx∗ (·) := |x (·)|}x∗∈X∗ is weak topology σ(X,X ) on X from Example 13.8. 13.13. Definition. Let X, τ  be a LCTVsp. Let {x } be a net in X. {ρα}α∈A β β∈B •{xβ} is Cauchy if and only if [∀ε > 0 , ∀α ∈ A, ∃βε,α ∈ B such that if β, γ > βε,α then ρα(xβ − xγ) < ε . • X is complete if and only if [each Cauchy net in X converges to a point in X].

13.14. Proposition. Let {ρα}α∈A and {dβ}β∈B be 2 families of seminorms on X. Then the following are equivalent. (1) {ρα}α∈A and {dβ}β∈B are equivalent families of seminorms, which by definition

means that τ{ρα}α∈A = τ{dβ }β∈B .

(2) ∀α ∈ A , ρα is τ{dβ }β∈B -continuous and

∀β ∈ B , dβ is τ{ρα}α∈A -continuous .

(3) ∀α ∈ A , ∃β1, . . . , βn ∈ B , ∃C > 0 , such that ∀x ∈ X , ρα(x) ≤ C(dβ1 (x) + ... + dβn (x))

and ∀β ∈ B , ∃α1, . . . , αn ∈ A , ∃D > 0 , such that ∀x ∈ X , dβ(x) ≤ D(ρα1 (x) + ... + ραn (x)) .

13.15. Definition. A family {ρα}α∈A of seminorms on X is directed provided

∀α, β ∈ A, ∃γ ∈ A, ∃C > 0 , such that ∀x ∈ X : ρα(x) + ρβ(x) ≤ Cργ(x) . Note that, by induction, this is equivalent to

∀α1, . . . , αn ∈ A, ∃γ ∈ A, ∃D > 0 , such that ∀x ∈ X : ρα1 (x) + ... + ραn (x) ≤ Dργ(x) .

13.16. Remark. Let {ρα}α∈A be a family of seminorms on X. Define •B = the set of all finite subsets of A P • for F ∈ B define dF := α∈F ρα. Then •{dF }F ∈B is a directed family of seminorms on X

• τ{ρα}α∈A = τ{dF }F ∈B . Recall. A T from a normed space X to a normed space Y is continuous if and only if there exists C > 0 so that kT xkY ≤ C kxkX for each x ∈ X. The next proposition is the corresponding statement for maps between spaces with topologies generated by seminorms.   13.17. Proposition. Let T : X, τ{ρα}α∈A → Y, τ{dβ }β∈B be a linear map. The following are equivalent. (1) T is continuous

(2) ∀β ∈ B , ∃α1, . . . , αn ∈ A , ∃C1 > 0 such that ∀x ∈ X : dβ(T x) ≤ C1 (ρα1 (x) + . . . ραn (x)) (3) ∀β ∈ B , ∃ a continuous seminorm ρ on X and ∃C2 > 0 such that ∀x ∈ X : dβ(T x) ≤ C2 ρ(x) And if {ρα}α∈A is a directed set, then also (4) ∀β ∈ B , ∃α ∈ A , ∃C3 > 0 such that ∀x ∈ X : dβ(T x) ≤ C1 ρα(x) . 13.18. Remark. We will often use Proposition 13.17 in the case of the formal identity j : X, τ  → X, τ . In this case, j is continuous if and only if ∃α , . . . , α ∈ {ρα}α∈A k·kX 1 n A , ∃C > 0 such that ∀x ∈ X : kT xkX ≤ C (ρα1 (x) + . . . ραn (x)). yr.mn.dy: 15.01.12 Page 3 of 4 Prof. Girardi 13. A Review of some Topology

13.19. Proposition. Let X be a LCTVsp with a natural topology τ generated by a family of seminorms. The following are equivalent. (1) (X, τ) is metrizable, i.e., ∃ a metric d: X × X → [0, ∞) such that τ equals the topology generated by the metric d. (2) ∃ a countable family {ρn}n∈N of seminorms on X that separates points such that τ = τ . {ρn}n∈N To show that (2) ⇒ (1), let   X ρn(x − y) d(x, y) := 2−n . (1) 1 + ρn(x − y) n∈N

In which case, it easy to see that if {xα}α∈A is a net in X, then

[{xα}α∈A is Cauchy in the metric d in (1)] ⇔ [{xα}α∈A is Cauchy for each seminorm ρn] . Thus a metrizable LCTVsp is complete as a metric space if and only if it is complete as a LCTVsp. 13.20. Definition. 1 A Fr´echetspace is a complete metrizable LCTVsp. 13.21. Theorem (Open Mapping Theorem). Let X and Y be Fr´echetspaces. Let f : X → Y be a continuous linear surjective mapping. Then f is open, i.e., if A is an open set in X then f(A) is an open set in Y . 13.22. Corollary. Let X and Y be Fr´echetspaces. Let f : X → Y be a continuous linear bijective mapping. Then f is a topological homeomorphism, i.e., f is bijective and f and f −1 are continuous. ∗ 13.23. Theorem. Let X be a Fr´echetspace. Let f, fn ∈ X be so that {fn}n∈N converges to ∗ f in the σ(X ,X)-topology. Then {fn}n∈N converges to f uniformly on compact subsets of X, i.e.,

∀ compact D ⊂ X, ∀ε > 0 , ∃N ∈ N , such that ∀n ≥ N, sup |fn(x) − f(x)| < ε . x∈D

References [RS-I] Michael Reed and Barry Simon, Methods of modern mathematical physics I, Second edition. Academic Press Inc., New York, 1980.

1This definition varies from book to book. yr.mn.dy: 15.01.12 Page 4 of 4