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 = |a|kxk,
I kx + yk ≤ kxk + kyk, for each x, y ∈ E and a ∈ 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 < ∞, let
N P∞ p lp = {(xn) ∈ R | n=0 |xn| < ∞}
and define a norm by
P∞ p 1/p k(xn)k = ( n=0 |xn| ) .
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 ∈ E and a ∈ 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 ) = {x ∈ E | Tx = 0}. Bounded linear mappings
Definition 5 A linear mapping T between normed spaces E and F is bounded if
T (BE ) = {Tx | x ∈ BE }
is bounded, where BE = {x ∈ E | kxk < 1}. 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 = sup{kTxk | x ∈ BE }
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 = sup{kTxk | x ∈ BE }
exists. Normable linear mappings
Proposition 8
If every bounded linear functional on l2 is normable, then
LPO: ∀α[∃n(α(n) 6= 0) ∨ ¬∃n(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
∞ 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 ∃n(α(n) 6= 0); in the latter case, we have ¬∃n(α(n) 6= 0). Located sets
Definition 9 A subset S of a metric space X is located if
d(x, S) = inf{d(x, y) | y ∈ S}
exists for each x ∈ 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 ∈ 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: ∀a ∈ 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 = |x| + |y|. Then there exists a bounded linear functional f such that f (1, a) = 1 + |a| and kf k = 1. Since |f (1, 0)| ≤ 1 and |f (0, 1)| ≤ 1, we have
1 + |a| = f (1, a) = f (1, 0) + af (0, 1) ≤ f (1, 0) + |a|,
and therefore f (1, 0) = 1 and af (0, 1) = |a|. 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 = {y − x | x ∈ A, y ∈ B}
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 ∈ A and y ∈ 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 ≡ {0} and B ≡ {x0}. 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 ∈ M and
kgk ≤ kf k + . Gˆateauxdifferentiable norm
Definition 18 The norm of a normed space E is Gˆateauxdifferentiable at x ∈ E with the derivative f : E → R if for each y ∈ E with kyk = 1 and > 0 there exists δ > 0 such that
∀t ∈ R(|t| < δ → |kx + tyk − kxk − tf (y)| ≤ |t|).
Note that the derivative f is linear. Definition 19 The norm of a normed space E is Gˆateauxdifferentiable if it is Gˆateauxdifferentiable at each x ∈ E with kxk = 1. Remark 20 The norm of lp for 1 < p < ∞ and the norm of a Hilbert space are Gˆateauxdifferentiable 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 Gˆateauxdifferentiable 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 ∈ E with kxk = kyk = 1. Remark 23 lp for 1 < p < ∞ 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 ∈ 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 ∈ E there exists a unique z ∈ 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 Gˆateauxdifferentiable 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 ∈ 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 ∈ 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 Gˆateauxdifferentiable norm such that the algebraic difference B − A = {y − x | x ∈ A, y ∈ B} 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 ∈ A and y ∈ B. Proof. Let z ∈ B − A be such that kzk = d(0, B − A), and let f be the derivative of the norm at z. Then f (x) + d ≤ f (y) for each x ∈ A and y ∈ B. Convex and sublinear functions
Definition 28 A function p of a linear space E into R is convex if
p(λx + (1 − λ)y) ≤ λp(x) + (1 − λ)p(y)
for each x, y ∈ E and λ ∈ [0, 1]. Definition 29 A function p of a linear space E into R is sublinear if
I p(ax) = ap(x),
I p(x + y) ≤ p(x) + p(y) for each x, y ∈ E and a ∈ R with a ≥ 0.
Note that sublinear functions are convex. Subderivative
Definition 30
I A linear functional f on a normed space E is a subderivative of a convex function p : E → R at x ∈ E if
f (y − x) ≤ p(y) − p(x)
for each y ∈ E;
I a convex function p : E → R is subdifferentiable at x ∈ E if it has a subderivative at x. Gˆateauxderivative
Definition 31 Let E be a normed space. Then a convex function p : E → R is Gˆateauxdifferentiable at x ∈ E with the derivative f : E → R if for each y ∈ E with kyk = 1 and > 0 there exists δ > 0 such that
∀t ∈ R(|t| < δ → |p(x + ty) − p(x) − tf (y)| ≤ |t|).
Note that the derivative f is linear, and if p is continuous at x, then f is bounded. Lemma 32 Let E be a normed space. If a convex function p : E → R is Gˆateauxdifferentiable at x ∈ E, then it is subdifferentiable at x. Subdifferentiablity and separation theorem
Proposition 33 (I 2017) Let C be a located convex subset of a normed space E, and let x ∈ E be such that d = d(x, C) > 0. Then the following are equivalent.
I The convex function d(·, C) is subdifferentiable at x.
I There exists a normable linear functional f on E such that kf k = 1 and f (x) + d ≤ f (y) for each y ∈ C. Classical dominated extension theorem
Theorem 34 (classical, Rudin 1991) Let p be a sublinear function on a linear space E, let M be a subspace of E, and let f be a linear functional on M such that
f (x) ≤ p(x)
for each x ∈ M. Then there exists a linear functional g on E such that g(x) = f (x) for each x ∈ M and
g(y) ≤ p(y)
for each y ∈ E. Subdifferentiablity and dominated extension theorem
Proposition 35 (I 2017) Let x be an element of a normed space E, and let p : E → R be a sublinear function. Then the following are equivalent.
I p is subdifferentiable at x.
I There exists a linear functional f on E such that f (x) = p(x) and f (y) ≤ p(y) for each y ∈ E. The Baire theorem
Theorem 36 (classical)
If (Gn) is a sequence of closed subsets of a complete metric space ∞ X such that X = ∪n=0Gn, then there exists n such that Gn has an inhabited interior. Theorem 37 (constructive)
If (Un) is a sequence of open dense subsets of a complete metric ∞ space X , then ∩n=0Un is dense in X . Gˆateauxdifferentiability of convex functions
Theorem 38 (Mazur 1933, I 2017) Let E be a separable Banach space, and let p : E → R be a continuous convex function. Then p is Gˆateauxdifferentiable at each point of a dense subset of E. Corollaries
Theorem 39 (I 2017) Let A and B be convex subsets of a separable Banach space E such that the algebraic difference
B − A = {y − x | x ∈ A, y ∈ B}
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 ∈ A and y ∈ B. Corollaries
Proof. Let z ∈ E be such that kzk < and d(·, B − A) is Gˆateaux differentiable, and hence subdifferentiable at z, Then there exists a normable linear functional f on E such that kf k = 1 and
f (z) + d ≤ f (y − x)
for each x ∈ A and y ∈ B. Therefore
− + d ≤ f (z) + d ≤ f (y) − f (x)
for each x ∈ A and y ∈ B. Corollaries
Theorem 40 (I 2017) Let x be an element of a separable Banach space E, and let p : E → R be a continuous sublinear function. Then for each > 0 there exists a bounded linear functional f on E such that p(x) ≤ f (x) + and f (y) ≤ p(y) for each y ∈ E. Acknowledgment
The speaker thanks thanks the Japan Society for the Promotion of Science (JSPS), Core-to-Core Program (A. Advanced Research Networks) and Grant-in-Aid for Scientific Research (C) No.16K05251 for supporting the research. References
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