5. Open Mapping Theorem and related Banach space theorems

Flashcards covering key statements and definitions from Open Mapping Theorem, Closed Graph Theorem, Banach–Steinhaus (Uniform Boundedness), Hahn–Banach (real and complex), and the geometric forms (Minkowski functional and separation theorems).

What does the Open Mapping Theorem (Theorem 5.1) say for a continuous linear map f: E → F between Banach spaces?

For a continuous linear map f: E → F between Banach spaces, f is surjective if and only if f is an open mapping.

What is an open mapping in topological terms?

A mapping f: X → Y is open if the image of every open subset of X is an open subset of Y.

Proposition 5.2 (criterion for an open mapping)

A linear map f: E → F between normed spaces is open iff there exists r > 0 such that f(BE(0,1)) ⊇ BF(0,r).

Statement of Lemma 5.3 about bounded sets and f

Let E be Banach, F be a normed space, f ∈ L(E,F), ε ∈ (0,1), and A ⊆ F be bounded with A ⊆ f(BE(0,1)) + εA. Then A ⊆ (1/(1−ε)) f(BE(0,1)).

Corollary 5.5 (Banach Isomorphism Theorem)

If E and F are Banach spaces and f ∈ L(E,F) is bijective, then f is an isomorphism; in particular, the inverse map f^−1 is continuous.

Corollary 5.6 (norm equivalence criterion)

Let N1 and N2 be norms on a vector space E such that (E,N1) and (E,N2) are Banach, and N2 ≤ α N1 for some α > 0. Then N1 and N2 are equivalent norms.

Theorem 5.9 (Closed Graph Theorem)

Let E and F be Banach spaces and f: E → F be linear. Then f is continuous iff its graph G_f = { (x, f(x)) : x ∈ E } is closed in E × F.

Theorem 5.11 (Banach–Steinhaus, Uniform Boundedness)

Let E and F be normed spaces and A ⊆ L(E,F). If for every x ∈ E the set {f(x) : f ∈ A} is bounded in F and A is nonmeager on a subset B ⊆ E, then A is bounded in L(E,F). In particular, if B = E, A is uniformly bounded.

Corollary 5.12 (boundedness criterion for sets of operators)

A ⊆ L(E,F) is bounded iff for every x ∈ E the set {f(x) : f ∈ A} is bounded in F.

Corollary 5.13 (pointwise convergence implies boundedness)

If (fn) ⊆ L(E,F) satisfies that for every x ∈ E, the sequence fn(x) converges in F, then (f_n) is bounded in L(E,F); the limit functional is continuous.

Hahn–Banach, real case (Theorem 5.17)

Let p: E → R be sublinear. If f is a linear form on a subspace H ⊆ E with f(x) ≤ p(x) for all x ∈ H, then f has a linear extension fr: E → R with fr(x) ≤ p(x) for all x ∈ E.

Hahn–Banach, complex case (Theorem 5.18)

The complex version of Hahn–Banach follows by reducing to the real case: there exists an extension of a linear form with the same domination bound, yielding a complex-valued extension with |f_r(x)| ≤ p(x).

Hahn–Banach extension (Theorem 5.19)

Let E be a vector space, H ⊆ E a subspace, and f ∈ L(H,K) a nonzero linear form. Then there exists fr ∈ L(E,K) extending f with fr(x) ≤ p(x) for all x ∈ E (in particular, extending f with the same domination bound as on H).

Corollary 5.20 (two standard HB consequences)

(1) For any nonzero x ∈ E there exists a continuous linear functional f on E with f(x) = ∥x∥ and ∥f∥ = 1. (2) If f, a linear functional on E satisfy f(x) = f(y) for all f ∈ E′, then x = y.

Definition and basic fact: Minkowski functional P_C (Definition 5.25)

For an open, convex subset C ⊆ E with 0 ∈ C, define PC(x) = inf{ α > 0 : α⁻¹ x ∈ C }. Then PC is sublinear; there exists M > 0 with PC(x) ≤ M||x|| for all x; and C = { x ∈ E : PC(x) < 1 }.

Theorem 5.26 (First geometric HB version)

Let E be a real normed vector space and A, B ⊆ E nonempty, disjoint and convex with A open. Then there exists a nonzero continuous linear functional f and α ∈ R such that f(x) ≤ α ≤ f(y) for all x ∈ A, y ∈ B (i.e., a closed affine hyperplane separates A and B).

Theorem 5.27 (Second geometric HB version)

Let E be a real normed space and A,B ⊆ E nonempty, disjoint and convex with A closed and B compact. Then there exists a nonzero continuous linear functional f and α ∈ R such that f(x) ≤ α ≤ f(y) for all x ∈ A, y ∈ B (strict separation).

Lemma 5.28 (separation from an open convex set)

If C ⊆ E is open and convex and x0 ∉ C, there exists a nonzero continuous linear functional f on E such that f(x) < f(x0) for all x ∈ C (i.e., a separating hyperplane between x0 and C).

Lemma 5.29 (distance between a closed set and a compact set)

If A is closed and B is compact in a metric space E and A,B are disjoint, then d(A,B) > 0 (positive distance between A and B).

Remark 5.30 (finite-dimensional caveat)

In general, without additional assumptions, two nonempty disjoint convex sets in an infinite-dimensional space may not be separated by a closed affine hyperplane; this separation is always possible in finite-dimensional spaces.