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The Constructive Hahn-Banach Theorem, Revisited

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The Constructive Hahn-Banach Theorem, Revisited The constructive Hahn-Banach theorem, revisited Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan Workshop: Constructive Mathematics HIM Bonn, 6{10 August, 2018 The Hahn-Banach theorem The Hahn-Banach theorem is a central tool in functional analysis, which is named after the mathematicians Hans Hahn and Stefan Banach who proved it independently in the late 1920s. The theorem has several forms. I The continuous extension theorem which allows the extension of a bounded linear functional defined on a subspace of a normed space to the whole space; I The separation theorem which allows a separation of two convex subsets of a normed space by a hyperplane, and has numerous uses in convex analysis (geometry); I The dominated extension theorem which is the most general form of the theorem, and allows the extension of a linear functional defined on a subspace of a normed space, and dominated by a sublinear function, to the whole space. Normed spaces Definition 1 A normed space is a linear space E equipped with a norm k · k : E ! R such that I kxk = 0 $ x = 0, I kaxk = jajkxk, I kx + yk ≤ kxk + kyk, for each x; y 2 E and a 2 R. Note that a normed space E is a metric space with the metric d(x; y) = kx − yk: Definition 2 A Banach space is a normed space which is complete with respect to the metric. Examples For 1 ≤ p < 1, let N P1 p lp = f(xn) 2 R j n=0 jxnj < 1g and define a norm by P1 p 1=p k(xn)k = ( n=0 jxnj ) : Then lp is a (separable) Banach space. Linear mappings Definition 3 A mapping T between linear spaces E and F is linear if I T (ax) = aTx, I T (x + y) = Tx + Ty for each x; y 2 E and a 2 R. A linear functional f on a linear space E is a linear mapping from E into R. Definition 4 The kernel ker(T ) of a linear mapping T between linear spaces E and F is defined by ker(T ) = fx 2 E j Tx = 0g: Bounded linear mappings Definition 5 A linear mapping T between normed spaces E and F is bounded if T (BE ) = fTx j x 2 BE g is bounded, where BE = fx 2 E j kxk < 1g: Proposition 6 Let T be a linear mapping between normed spaces E and F . Then the following are equivalent. I T is continuous, I T is uniformly continuous, I T is bounded. Normable linear mappings Classicaly, for a bounded linear mapping T between normed spaces, its norm kT k = supfkTxk j x 2 BE g always exists, and hence the set E ∗ of all bounded linear fuctionals on a normed space E forms a Banach space. However it is not always the case constructively. Definition 7 A linear mapping T between normed spaces E and F is normable if kT k = supfkTxk j x 2 BE g exists. Normable linear mappings Proposition 8 If every bounded linear functional on l2 is normable, then LPO: 8α[9n(α(n) 6= 0) _ :9n(α(n) 6= 0)] holds. Proof. Let α be a binary sequence with at most one nonzero term, and define a linear functional f on l2 by 1 X f ((xn)) = α(k)xk : k=0 Then f is bounded. If f is normable, then either 0 < kf k or kf k < 1; in the former case, we have 9n(α(n) 6= 0); in the latter case, we have :9n(α(n) 6= 0). Located sets Definition 9 A subset S of a metric space X is located if d(x; S) = inffd(x; y) j y 2 Sg exists for each x 2 X . Normable linear functionals Proposition 10 (I 1992) Let T be a linear mapping of a normed space E ontoa finite-dimensional normed space F . Then the following are equivalent. I T (BE ) is located in F . I ker(T ) is located in E. Corollary 11 A nonzero bounded linear functional f on a normed space E is normable if and only if its kernel ker(f ) is located. Classical continuous extension theorem Theorem 12 (classical) Let M be a subspace of a normed space E, and let f be a bounded linear functional on M. Then there exists a bounded linear functional g on E such that g(x) = f (x) for each x 2 M and kgk = kf k. Corollary 13 (classical) Let x be a nonzero element of a normed space E. Then there exists a bounded linear functional f on E such that f (x) = kxk and kf k = 1. Classical continuous extension theorem Proposition 14 The corollary of classical continuous extension theorem implies LLPO: 8a 2 R(0 ≤ a _ a ≤ 0): Proof. Let (1; a) be a nonzero element of the normed space R2 with a norm k(x; y)k = jxj + jyj. Then there exists a bounded linear functional f such that f (1; a) = 1 + jaj and kf k = 1. Since jf (1; 0)j ≤ 1 and jf (0; 1)j ≤ 1, we have 1 + jaj = f (1; a) = f (1; 0) + af (0; 1) ≤ f (1; 0) + jaj; and therefore f (1; 0) = 1 and af (0; 1) = jaj. Either −1 < f (0; 1) or f (0; 1) < 1. In the former case, we have 0 ≤ a; in the latter case, we have a ≤ 0. The constructive separation theorem Theorem 15 (Bishop 1967) Let A and B be bounded convex subsets of a separable normed space E such that the algebraic difference B − A = fy − x j x 2 A; y 2 Bg is located and d = d(0; B − A) > 0. Then for each > 0 there exists a normable linear functional f on E such that kf k = 1 and f (x) + d ≤ f (y) + for each x 2 A and y 2 B. The corollary Corollary 16 (Bishop 1967) Let x0 be a nonzero element of a separable normed space E. Then for each > 0 there exists a normable linear functional f on E such that kf k = 1 and kx0k ≤ f (x0) + . Proof. Apply Theorem 15 for A ≡ f0g and B ≡ fx0g. The constructive continuous extension theorem Theorem 17 (Bishop 1967, Bridges-Richman 1987) Let M be a subspace of a separable normed space E, and let f be a nonzero linear functional on M such that the kernel ker(f ) is located in E. Then for each > 0 there exists a normable linear functional g on E such that g(x) = f (x) for each x 2 M and kgk ≤ kf k + . G^ateauxdifferentiable norm Definition 18 The norm of a normed space E is G^ateauxdifferentiable at x 2 E with the derivative f : E ! R if for each y 2 E with kyk = 1 and > 0 there exists δ > 0 such that 8t 2 R(jtj < δ ! jkx + tyk − kxk − tf (y)j ≤ jtj): Note that the derivative f is linear. Definition 19 The norm of a normed space E is G^ateauxdifferentiable if it is G^ateauxdifferentiable at each x 2 E with kxk = 1. Remark 20 The norm of lp for 1 < p < 1 and the norm of a Hilbert space are G^ateauxdifferentiable at each x with x # 0. A constructive corollary Proposition 21 (I 1989) Let x be a nonzero element of a normed linear space E whose norm is G^ateauxdifferentiable at x. Then there exists a unique normable linear functional f on E such that f (x) = kxk and kf k = 1. Proof. Take the derivative f of the norm at x. Uniformly convex spaces Definition 22 A normed space E is uniformly convex if for each > 0 there exists δ > 0 such that kx − yk ≥ ! k(x + y)=2k ≤ 1 − δ for each x; y 2 E with kxk = kyk = 1. Remark 23 lp for 1 < p < 1 and a Hilbert space are uniformly convex. Uniformly convex spaces Proposition 24 (Bishop-Bridges 1985) Let f be a nonzero normable linear functional on a uniformly convex Banach space E. Then there exists x 2 E such that f (x) = kf k and kxk = 1. Proposition 25 (Bishop 1967) Let C be a closed located convex subset of a uniformly convex Banach space E. Then for each x 2 E there exists a unique z 2 C such that kx − zk = d(x; C): A constructive continuous extension theorem Theorem 26 (I 1989) Let M be a subspace of a uniformly convex Banach space E with a G^ateauxdifferentiable norm, and let f be a normable linear functional on M. Then there exists a unique normable linear functional g on H such that g(x) = f (x) for each x 2 M and kgk = kf k. Proof. We may assume without loss of generality that kf k = 1. Let M be the closure of M. Then there exists a normable extension f of f on M. Since M is a uniformly convex Banach space, there exists x 2 M such that f (x) = kxk = 1: Take the derivative g of the norm at x. A constructive separation theorem Theorem 27 (I 1989) Let A and B be convex subsets of a uniformly convex Banach space E with a G^ateauxdifferentiable norm such that the algebraic difference B − A = fy − x j x 2 A; y 2 Bg is located and d = d(0; B − A) > 0. Then there exists a normable linear functional f on E such that kf k = 1 and f (x) + d ≤ f (y) for each x 2 A and y 2 B.
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