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 space (or equivalently τ is a topology on X) provided • if A; B 2 τ, then A \ B 2 τ • for any index set Γ, if fAαgα2Γ ⊂ τ then [α2ΓAα 2 τ •;;X 2 τ. In which case, A 2 P(X) is an open set if and only if A 2 τ. 13.2. Definition. Let (X; τ) be a topological space and B ⊂ τ. Then B is a base (or basis) of τ provided 8A 2 τ 9fBαgα2Γ ⊂ τ such that A = [α2ΓBα : 13.3. Example. Let X be a normed linear space. As usual, for " > 0 and x 2 X, let N"(x) := fy 2 X : kx − ykX < "g : Let τ be the open subsets of X, i.e., τ = fA 2 P(X): 8x 2 A; 9" > 0 such that N"(x) ⊂ Ag : Then a base B for (X; τ) is B := fN"(x): x 2 X; " > 0g : 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 vector space 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 2 X and x 6= y then there is a f 2 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) := ff : X ! K j f is continuous and linearg : 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 2 Z is continuous. • Clearly, each f 2 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 2 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 2 Z;Ui is open in K ; n 2 N : Another, more convenient, base for σ(X; Z) is n −1 \i=1fi [N" (fi (x0))] : fi 2 Z ; " > 0 ; x0 2 X ; n 2 N ; note that n −1 \i=1fi [N" (fi (x0))] = fx 2 X : jfi(x) − fi(x0)j < " ; i = 1; : : : ; ng = fx 2 X : jfi(x − x0)j < " ; i = 1; : : : ; ng =: N";f1;:::;fn (x0) : Since Z separates points, σ(X; Z) is a Hausdorff (i.e., T2) topology, i.e., if x1; x2 2 X and x1 6= x2 then there are disjoint U1;U2 2 σ(X; Z) such that xi 2 Ui. Thus, if the limit of a net 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 Banach space over K, endowed with a norm k·kX and the norm topology τ as in Example 13.3. Let k·kX X∗ := X; τ ∗ = fx∗ : X ! j x∗ is linear and (norm) continuousg : k·kX K (1) 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 ∗ N ∗ ∗ (x) = fy 2 X : jx (x − y)j < " ; for i = 1; : : : ; ng : ";x1;:::;xn i ∗ ∗ ∗ where x 2 X and x1; : : : ; xn 2 X and " > 0 and n 2 N. ∗ ∗ ∗ " Note σ(X; X ) ⊂ τk·k since N";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 ∗ ∗ ∗ ∗ ∗ N";x1:::;xn (x ) = fy 2 X : j(x − y )(xi)j < " ; for i = 1; : : : ; ng : ∗ ∗ where x 2 X and x1; : : : ; xn 2 X and " > 0 and n 2 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 2 X such that x∗∗(x∗) = x∗(x) for each x∗ 2 X∗. 13.9. Optional Fact. Let X be a normed linear space. Then the following are equivalent. • dim X < 1. • (X; σ (X; X∗)) is metrizable. • (X∗; σ (X∗;X)) is metrizable. Thus, often sequences 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; 1) be a function. Then ρ is a seminorm (in German, Halbnorm) provided 8x; y 2 X and α 2 K: • ρ(x + y) ≤ ρ(x) + ρ(y) • ρ(α x) = jαj 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 fραgα2A of seminorms separates points provided if ρα(x) = 0 for each α 2 A then x = 0. 13.11. Definition. A space X is called a locally convex topological vector space (i.e. (LCTVsp)) provided (1) X is a vector space over K (2) 9 a family fραgα2A of seminorms on X that separates points. In this case: (3) the natural topology τ = τfραgα2A on X, generated by the seminorms fραgα2A, is the weakest topology so that: • ρα is continuous for each α 2 A • the mapping +: X × X ! X, defined by (x; y) ! x + y, is continuous . Note that since fραgα2A 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 fNε,α1,...,αn (x) j αi 2 A ; " > 0 ; n 2 N ; x 2 Xg where Nε,α1,...,αn (x) := fy 2 X : ραi (x − y) < " ; i 2 f1; : : : ; ngg, which is a convex set (5) if fxβgβ2B is a net in X, then xβ ! x () ρα(xβ − x) ! 0 8α 2 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. ∗ ∗ • fρx∗ (·) := jx (·)jgx∗2X∗ is weak topology σ(X; X ) on X from Example 13.8. 13.13. Definition. Let X; τ be a LCTVsp. Let fx g be a net in X. fραgα2A β β2B • fxβg is Cauchy if and only if [8" > 0 ; 8α 2 A; 9βε,α 2 B such that if β; γ > βε,α then ρα(xβ − xγ) < " : • X is complete if and only if [each Cauchy net in X converges to a point in X]. 13.14. Proposition. Let fραgα2A and fdβgβ2B be 2 families of seminorms on X. Then the following are equivalent. (1) fραgα2A and fdβgβ2B are equivalent families of seminorms, which by definition means that τfραgα2A = τfdβ gβ2B . (2) 8α 2 A , ρα is τfdβ gβ2B -continuous and 8β 2 B , dβ is τfραgα2A -continuous . (3) 8α 2 A , 9β1; : : : ; βn 2 B , 9C > 0 , such that 8x 2 X , ρα(x) ≤ C(dβ1 (x) + ::: + dβn (x)) and 8β 2 B , 9α1; : : : ; αn 2 A , 9D > 0 , such that 8x 2 X , dβ(x) ≤ D(ρα1 (x) + ::: + ραn (x)) . 13.15. Definition. A family fραgα2A of seminorms on X is directed provided 8α; β 2 A; 9γ 2 A; 9C > 0 ; such that 8x 2 X : ρα(x) + ρβ(x) ≤ Cργ(x) : Note that, by induction, this is equivalent to 8α1; : : : ; αn 2 A; 9γ 2 A; 9D > 0 ; such that 8x 2 X : ρα1 (x) + ::: + ραn (x) ≤ Dργ(x) : 13.16. Remark. Let fραgα2A be a family of seminorms on X. Define •B = the set of all finite subsets of A P • for F 2 B define dF := α2F ρα. Then • fdF gF 2B is a directed family of seminorms on X • τfραgα2A = τfdF gF 2B . Recall. A linear map 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 2 X. The next proposition is the corresponding statement for maps between spaces with topologies generated by seminorms. 13.17. Proposition. Let T : X; τfραgα2A ! Y; τfdβ gβ2B be a linear map. The following are equivalent. (1) T is continuous (2) 8β 2 B , 9α1; : : : ; αn 2 A , 9C1 > 0 such that 8x 2 X : dβ(T x) ≤ C1 (ρα1 (x) + : : : ραn (x)) (3) 8β 2 B , 9 a continuous seminorm ρ on X and 9C2 > 0 such that 8x 2 X : dβ(T x) ≤ C2 ρ(x) And if fραgα2A is a directed set, then also (4) 8β 2 B , 9α 2 A , 9C3 > 0 such that 8x 2 X : dβ(T x) ≤ C1 ρα(x) .