From Hahn-Banach to Monotonicity by Stephen Simons

By Stephen Simons

In this re-creation of LNM 1693 the fundamental suggestion is to minimize questions about monotone multifunctions to questions about convex services. although, instead of utilizing a “big convexification” of the graph of the multifunction and the “minimax technique”for proving the lifestyles of linear functionals pleasing yes stipulations, the Fitzpatrick functionality is used. the adventure starts off with a generalization of the Hahn-Banach theorem uniting classical useful research, minimax conception, Lagrange multiplier concept and convex research and culminates in a survey of present effects on monotone multifunctions on a Banach space.

The first chapters are geared toward scholars drawn to the improvement of the fundamental theorems of useful research, which leads painlessly to the idea of minimax theorems, convex Lagrange multiplier idea and convex research. the remainder 5 chapters are precious if you desire to find out about the present study on monotone multifunctions on (possibly non reflexive) Banach space.

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If D is a subspace of a normed space E and z ∗ ∈ E ∗ and y, z ∗ = 0 for all y ∈ D =⇒ z ∗ = 0, then D is dense in E. 7. 4, with D := F − C. If F is a nonempty closed convex subset of a normed space E, C is a nonempty w(E, E ∗ )–compact convex subset of E and F ∩ C = ∅ then there exists z ∗ ∈ E ∗ such that supF z ∗ < inf C z ∗ . 4, pp. 58–59]. We conclude this section by mentioning James’s theorem, one of the most beautiful results in functional analysis: if C is a nonempty bounded closed convex subset of a Banach space E then C is w(E, E ∗ )–compact if, and only if, for all x∗ ∈ E ∗ , there exists x ∈ C such that x, x∗ = maxC x∗ .

So we can suppose that β ∈ R. 5. 2, there exists a linear functional L on E such that L ≤ Q. Since Q ≤ P , L ≤ P , as required. Let d ∈ D. Then L(d) = −L −d) ≥ −Q(−d) ≥ β. Taking the infimum over d ∈ D, inf D L ≥ β = inf D P. On the other hand, since L ≤ P , inf D L ≤ inf D P. 7. 6 is “forced” in the sense that if L is linear, L ≤ P and β = inf D P = inf D L ∈ R then, as the reader can easily verify, L ≤ Q. 4 that any sublinear functional is the pointwise supremum of the linear functionals that it dominates.

Thus if inf D P ∈ R then Q is the pointwise supremum of the linear functionals L such that L ≤ P and inf D L = inf D P . 20 I The Hahn-Banach-Lagrange theorem and some consequences Let X be a nonempty convex subset of a vector space, and f : X → R. We say that f is convex if x, y ∈ X and λ ∈ ]0, 1[ =⇒ f λx + (1 − λ)y ≤ λf (x) + (1 − λ)f (y). We say that f is concave if x, y ∈ X and λ ∈ ]0, 1[ =⇒ f λx + (1 − λ)y ≥ λf (x) + (1 − λ)f (y). 8. If X is a nonempty set and f : X → ]−∞, ∞], we write dom f := x ∈ X: f (x) ∈ R .

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