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B4a Banach spaces Michaelmas 2010 lectured by Bernd Kirchheim based on notes by CJK Batty
Lecture 15
5.Corollary The Hahn-Banach Theorem is true for every separable real normed space (X, ‖ · ‖):
Y subspace of X, g ∈ Y ′^ ⇒ ∃G ∈ X′^ : G|Y = g and ‖G‖X′ = ‖g‖Y ′.
6.Proposition Let (X, ‖ · ‖) be a complex normed space. We con- sider the real linear space XR obtained from X when allowing multiplica- tion only by α ∈ R, the (XR, ‖ · ‖) is a real normed space. Moreover, if the Hahn-Banach Theorem holds true for (XR, ‖ · ‖) then it holds also for (X, ‖ · ‖). In particular, we have proven it for all separable complex spaces.
Seperation of convex sets
7.Proposition Let (X, ‖ · ‖) be a separable real normed space and let 0 ∈ U ⊂ X be open and convex and xo ∈/ U. Then there is
G ∈ X′^ such that G(y) < G(x 0 ) for all y ∈ U.
- Corollary Let (X, ‖·‖) be a separable real normed space, and V, W ⊂ X be convex. If V is open and V ∩ W = ∅ then there exists g ∈ X′^ and α ∈ R such that
for all v ∈ V, w ∈ W : g(v) < α ≤ g(w). (α = inf g(W ))
- Corollary Let (X, ‖ · ‖) be a separable real normed space, W ⊂ X closed and convex and x /∈ W. Then there is g ∈ X′^ and α ∈ R such that
g(x) < α < g(w) for all w ∈ W.
Proof As W is closed there is V = Bε(x) open and convex with V ∩W = ∅, now apply Corollary 8. 10.Proposition Existence of a subdifferential Let (X, ‖ · ‖) be a separable real normed space and f : X → R be convex and continuous (∀t : {f < t} open is enough). Then for all x ∈ X there is G ∈ X′^ such that
f (y) ≥ f (x) + G(y − x) for all y ∈ X. (1)
11.Corollary Let (X, ‖ · ‖) be a separable real normed space and f : X → R be convex and continuous. If f has in x a local minimum, then this minimum is global. 12.Practicing Exercise Let (X, ‖ · ‖) be a real normed space, and con- sider the following two staments. Show that the following two statements are equivalent, giving direct proofs (without going through all of Prop2, Cor 3 and Lemma4)
a) Hahn-Banch Theorem(analytic version): For all F : X → R convex continuous, any subspace Y and g ∈ Y ′^ with g ≤ F|Y there is G ∈ X′ s.t. G ≤ F and G|Y = g.
b) Hahn-Banch Theorem(geometric version) Let V, W be convex sets. If V is open and V ∩ W = ∅ then there exists a bounded linear functional and α ∈ R such that for all v ∈ V, w ∈ W : g(v) < α ≤ g(w).
Show directly, i.e without going through all of Prop 2, Cor 3 and Lemma 4, that a) implies b) and that if b) holds for X × R then a) holds for X. [Hint: a) implies b) uses Prop 7 and Cor 8. Showing b) implies a) is more interesting, look at the open epigraph {(x, t) : F (x) < t} and the graph of g.]
8 Invertibility and spectrum
Let T : X → Y be a bounded linear operator between normed vector spaces X and Y. We say that T is invertible if there exists a bounded linear operator S : Y → X such that ST = IX and T S = IY ; then we write S = T −^1 , the inverse of T. Note that in infinite-dimensional spaces these two equalities are independent of one another (see Example 6.20 with left and right shift T ′T = Il 2 6 = T T ′). If T : X → Y is a bijection, then there is an algebraic inverse. In fact, if X and Y are Banach spaces and T : X → Y is a continuous linear bijection, then the algebraic inverse is automatically bounded! The proof of this, however, is beyond the scope of this course only in 4th year. Nevertheless, we shall assume in this section that X and Y are Banach spaces, and Y = X. There is really no loss in doing so, since an invertible operator extends to an invertible operator on the completions (see Theorem 5.8). Our aim is to generalize the notion/theory of eigenvalues of a T ∈ B(X) from linear finite dimensional to Banach spaces, so to investigate when T − λI is invertible.
