On Topological Dual Of A NUCLEARAND CO-NUCLEAR SPACES
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ON TOPOLOGICAL DUAL OF A NUCLEARAND CO-NUCLEAR SPACES S. N. Sah, Ph.D.Scholar, Department of Mathematics, Patna University, Patna, Bihar, India. E-mail: [email protected] Abstract: This paper is intended to discuss the some interesting properties of Nuclear and Co-nuclear spaces. It is shown that there are some nuclear spaces which are not co-nuclear and co-nuclear spaces which are not nuclear. Keywords: Nuclear, Co-nuclear, Strong dual and Infrabarrelled. 1. INTRODUCTION: The concept of a nuclear space arose in an investigation of the question: For what spaces are the analogous Schwarz’s kernel theorem valid? The fundamental results in the theory of nuclear spaces are due to A. Grothendieck [1]. The function spaces used in analysis are, as a rule, Banach or nuclear spaces. Nuclear space play an important role in the spectral analysis of operators on Hilbert spaces(the construction of rigged Hilbert spaces, expansions in terms of generalized eigen vectors etc.) Nuclear spaces are closely connected with measure theory on locally convex spaces[2]. The definition of nuclear locally convex spaces which was introduced by A. Grothendieck in 1951 in setting ofhis theory of topological tensor products[1]. G. Köthe has shown that every complete quasi-barreled co-nuclear space has the approximation property. He also asked whether the same is true for any co-nuclear space[3]. In this paper,we discuss some interesting properties of nuclear space and co-nuclear space. Let E be a locally convex space with dual E ' . Then E ' is a vector subspace of the algebraic dual E ¿ of E [5]. Also to each element x of E corresponds a linear form x(f) = f(x). The mapping of E into E ' ¿ thus defined is clearly linear; if E is separated it is also (1,1) for x = y iff f(x) = f(y) for all f ∈E ' , and by a corollary of Hahn-Banach theorem this is equivalent to x = y. Thus E is identified with a vector subspace Eof E ' ¿ . We shall see that this algebraic symmetry between E and E ' , in which each is (isomorphic to) a vector subspace of the algebraic dual of the other, extends to a topological one; there are topologies on E ' under which it is a separated convex space with (continuous) dual E[7,8]. Page 1 of 5
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A NUCLEARAND CO-NUCLEAR SPACES
Transcript of On Topological Dual Of A NUCLEARAND CO-NUCLEAR SPACES
ON TOPOLOGICAL DUAL OF A NUCLEARAND CO-NUCLEAR SPACES S. N. Sah, Ph.D.Scholar, Department of Mathematics, Patna University, Patna, Bihar, India.E-mail: [email protected]:This paper is intended to discuss the some interesting properties of Nuclear and Co-nuclear spaces. It is shown that there are some nuclear spaces which are not co-nuclear and co-nuclear spaces which are not nuclear.Keywords: Nuclear, Co-nuclear, Strong dual and Infrabarrelled.1. INTRODUCTION: The concept of a nuclear space arose in an investigation of the question: For what spaces are the analogous Schwarzs kernel theorem valid? The fundamental results in the theory of nuclear spaces are due to A. Grothendieck [1]. The function spaces used in analysis are, as a rule, Banach or nuclear spaces. Nuclear space play an important role in the spectral analysis of operators on Hilbert spaces(the construction of rigged Hilbert spaces, expansions in terms of generalized eigen vectors etc.) Nuclear spaces are closely connected with measure theory on locally convex spaces[2].The definition of nuclear locally convex spaces which was introduced by A. Grothendieck in 1951 in setting ofhis theory of topological tensor products[1].G. Kthe has shown that every complete quasi-barreled co-nuclear space has the approximation property. He also asked whether the same is true for any co-nuclear space[3].In this paper,we discuss some interesting properties of nuclear space and co-nuclear space.Let E be a locally convex space with dual . Then is a vector subspace of the algebraic dual of E [5]. Also to each element x of E corresponds a linear form (f) = f(x). The mapping of E into thus defined is clearly linear; if E is separated it is also (1,1) for = iff f(x) = f(y) for all f , and by a corollary of Hahn-Banach theorem this is equivalent to x = y. Thus E is identified with a vector subspace of . We shall see that this algebraic symmetry between E and , in which each is (isomorphic to) a vector subspace of the algebraic dual of the other, extends to a topological one; there are topologies on under which it is a separated convex space with (continuous) dual E[7,8].We denote the elements of E' by x', y', and write < x, x' > for the value of the linear form x' at the point x of E. Then 0.(It follows from a corollary of Hahn-Banach theorem).(D') For each x' 0 in E, there exists x E with < x, x' > 0. More generally, let E and E' be any two vector spaces over the same (real or complex) scalar field, and let and
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