منابع مشابه
Hahn-banach Theorem
We prove a version of Hahn-Banach Theorem. and 1] provide the notation and terminology for this paper. The following propositions are true: (1) For all sets x, y and for every function f such that h hx; yi i 2 f holds y 2 rng f: (2) For every set X and for all functions f, g such that X dom f and f g holds fX = gX: (3) For every non empty set A and for every set b such that A 6 = fbg there exis...
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(2)1 For every set X and for all functions f , g such that X ⊆ dom f and f ⊆ g holds f X = g X . (3) For every non empty set A and for every set b such that A 6= {b} there exists an element a of A such that a 6= b. (4) For all sets X , Y holds every non empty subset of X→̇Y is a non empty functional set. (5) Let B be a non empty functional set and f be a function. Suppose f = ⋃ B. Then dom f = ⋃...
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The idea behind this article is to provide a unified and relatively nontechnical framework for treating the main existence theorems for continuous linear functionals in functional analysis, convex analysis, Lagrange multiplier theory, and minimax theory. While many of the results in this article are already known, our approach is new, and gives a large number of results with considerable econom...
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Let T̃ = ∑n i=1 ũi⊗ vi : V → V = [v1, ..., vn] ⊂ X, where ũi ∈ V ∗ and X is a Banach space. Let T = ∑n i=1 ui ⊗ vi : X → V be an extension of T̃ to all of X (i.e., ui ∈ X∗) such that T has minimal (operator) norm. In this paper we show in particular that, in the case n = 2 and the field is R, there exists a rank-n T̃ such that ‖T‖ = ‖T̃‖ for all X if and only if the unit ball of V is either not smo...
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was established by Hahn [1; p. 217] in 1927, and independently by Banach [2; p. 212] in 1929, who also generalized Theorem 0 for real spaces, to the situation in which the functional q :E^>R is an arbitrary subadditive, positive homogeneous functional [2; p. 226]. Theorem 0 was not established for complex spaces until 1938, when it was deduced from the real theorem by Bohnenblust and Sobczyk [3...
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ژورنال
عنوان ژورنال: Mathematical Proceedings of the Cambridge Philosophical Society
سال: 2006
ISSN: 0305-0041,1469-8064
DOI: 10.1017/s0305004105008935