Bidual Spaces and Reflexivity of Real Normed Spaces

In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].Narita Keiko - Hirosaki-city Aomori, JapanEndou Noboru - Gifu National College of Technology Gifu, JapanShidama Yasunari - Shinshu University Nagano, JapanHaim Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Peter D. Dax. Functional Analysis. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley Interscience, 2002.Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213-219, 2006. doi:10.2478/v10037-006-0024-x.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35-40, 1990.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Topological properties of real normed space. Formalized Mathematics, 22(3):209-223, 2014. doi:10.2478/forma-2014-0024.Keiko Narita, Noboru Endou, and Yasunari Shidama. Dual spaces and Hahn-Banach theorem. Formalized Mathematics, 22(1):69-77, 2014. doi:10.2478/forma-2014-0007.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29-34, 1993.Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Michael Reed and Barry Simon. Methods of modern mathematical physics. Vol. 1. Academic Press, New York, 1972.Walter Rudin. Functional Analysis. New York, McGraw-Hill, 2nd edition, 1991.Hideki Sakurai, Hisayoshi Kunimune, and Yasunari Shidama. Uniform boundedness principle. Formalized Mathematics, 16(1):19-21, 2008. doi:10.2478/v10037-008-0003-5.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377-380, 2003.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec. Subspaces of real linear space generated by one, two, or three vectors and their cosets. Formalized Mathematics, 3(2):271-274, 1992.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Geometry of integral polynomials, MM-ideals and unique norm preserving extensions

We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space $X$ is $\{\pm \phi^k: \phi \in X^*, \| \phi\|=1\}$. With this description we show that, for real Banach spaces $X$ and $Y$, if $X$ is a non trivial $M$-ideal in $Y$, then $\hat\bigotimes^{k,s}_{\epsilon_{k,s}} X$ (the $k$-th symmetric tensor product of $X$ endowed with the injective symmetric tensor norm) is \emph{never} an $M$-ideal in $\hat\bigotimes^{k,s}_{\epsilon_{k,s}} Y$. This result marks up a difference with the behavior of non-symmetric tensors since, when $X$ is an $M$-ideal in $Y$, it is known that $\hat\bigotimes^k_{\epsilon_k} X$ (the $k$-th tensor product of $X$ endowed with the injective tensor norm) is an $M$-ideal in $\hat\bigotimes^k_{\epsilon_k} Y$. Nevertheless, if $X$ is Asplund, we prove that every integral $k$-homogeneous polynomial in $X$ has a unique extension to $Y$ that preserves the integral norm. We explicitly describe this extension. We also give necessary and sufficient conditions (related with the continuity of the Aron-Berner extension morphism) for a fixed $k$-homogeneous polynomial $P$ belonging to a maximal polynomial ideal \Q(^kX) to have a unique norm preserving extension to \Q(^kX^{**}). To this end, we study the relationship between the bidual of the symmetric tensor product of a Banach space and the symmetric tensor product of its bidual and show (in the presence of the BAP) that both spaces have `the same local structure'. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.Comment: 25 page

Helly's selection theorem and the principle of local reflexivity of ordered type

AbstractLet (E,E+,∥ · ∥) be an ordered normed space with a positive cone E+, let 0 ≤ ψ ϵ E″, let N be finite-dimensional subspace of E′ and ε > 0. In terms of the notions of half-full injections and half-decomposable surjections, sufficient conditions for N to ensure the existence of x ϵ E+ with ∥x∥≤∥ψ∥ + ϵ and ψ=KEx on N have been found (Theorems 3.5 and 3.6). As an application of Helly's selection theorem of ordered type, the principle of local reflexivity of ordered type is obtained (Theorem 4.7)

Vector-valued invariant means revisited

AbstractWe show that a Banach space X is complemented in its ultraproducts if and only if for every amenable semigroup S the space of bounded X-valued functions defined on S admits (a) an invariant average; or (b) what we shall call “an admissible assignment”. Condition (b) still provides an equivalence for quasi-Banach spaces, while condition (a) necessarily implies that the space is locally convex