Sequences and Series in Banach Spaces

Diestel, Joe

doi:10.1007/978-1-4612-5200-9

Cited by 1,109 publications

(639 citation statements)

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“…We will define a space by choosing a norm and let the space consist of the sequences that have finite norm as is common in Banach space theory. If the norm makes the space complete it is called a Banach sequence space [Die84]. Interesting examples are ℓ ∞ of bounded sequences with the maximum norm ∥(α k )∥ ∞ = max |α k |, c 0 of sequence that converges to 0 equipped with the same maximum norm and ℓ p which for 1 ≤ p < ∞ is defined by the norm…”

Section: Countable Sets Of Eventsmentioning

confidence: 99%

Axioms for Rational Reinforcement Learning

Sunehag

Hutter

2011

Lecture Notes in Computer Science

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We provide a formal, simple and intuitive theory of rational decision making including sequential decisions that affect the environment. The theory has a geometric flavor, which makes the arguments easy to visualize and understand. Our theory is for complete decision makers, which means that they have a complete set of preferences. Our main result shows that a complete rational decision maker implicitly has a probabilistic model of the environment. We have a countable version of this result that brings light on the issue of countable vs finite additivity by showing how it depends on the geometry of the space which we have preferences over. This is achieved through fruitfully connecting rationality with the Hahn-Banach Theorem. The theory presented here can be viewed as a formalization and extension of the betting odds approach to probability of Ramsey and De Finetti [Ram31,deF37].

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Section: Countable Sets Of Eventsmentioning

confidence: 99%

Axioms for Rational Reinforcement Learning

Sunehag

Hutter

2011

Lecture Notes in Computer Science

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“…When X is infinite dimensional, then by the Riesz Lemma [12] there exists a sequence {y n } such that y n = 1 for n=1,2,... and for n = 1, 2, ...…”

Section: Theorem 31 I) An (X) = ∞ If and Only If (X · ) Is Finite mentioning

confidence: 99%

“…ii) If (X, · ) is a Schur space [12] , then the following equality sup inf r a y, {x n i } i≥1 : {x n i } i≥1 is weakly convergent and…”

Section: Theorem 31 I) An (X) = ∞ If and Only If (X · ) Is Finite mentioning

confidence: 99%

Uniform asymptotic normal structure, the uniform semi‐Opial property and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part I

Budzyńska¹,

Kuczumow²,

Reich

1998

Abstract and Applied Analysis

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“…Therefore K is weakly sequentially compact in .el. By Schur's Theorem (see Diestel [6], page 85, and Corollary 14 page 296 of Dunford-Schwartz [7]), weak convergence and norm convergence of sequences are equivalent in .el; hence K is norm-compact in .e1.0 whenever Sn /' S or Sn '\. S. Note that our definition of continuity coincides with that of Schmeidler [16] (see also Aumann and Shapley [2]).…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

The least core, kernel and bargaining sets of large games

1998

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_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __We study the least core, the kernel, and bargaining sets of coalitional games with a countable set of players. We show that the least core of a continuous superadditive game with a countable set of players is a non-empty (norm-compact) subset of the space of all countable additive measures. Then we show that in such games the intersection of the prekernel and least core is non-empty. Finally, we show that this intersection is contained in the Aumann-Maschler and the Mas-Colell bargaining sets.

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Sequences and Series in Banach Spaces

Cited by 1,109 publications

References 0 publications

Axioms for Rational Reinforcement Learning

Axioms for Rational Reinforcement Learning

Uniform asymptotic normal structure, the uniform semi‐Opial property and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part I

The least core, kernel and bargaining sets of large games

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