Multi-normed spaces

H. G. Dales. Multi-normed spaces. 2012. <http://eudml.org/doc/285986>.

@book{H2012,
abstract = {We modify the very well known theory of normed spaces (E,||·||) within functional analysis by considering a sequence (||·||ₙ: n ∈ ℕ) of norms, where ||·||ₙ is defined on the product space Eⁿ for each n ∈ ℕ. Our theory is analogous to, but distinct from, an existing theory of ’operator spaces’; it is designed to relate to general spaces $L^\{p\}$ for p ∈ [1,∞], and in particular to L¹-spaces, rather than to L²-spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory, that we shall use, we shall present in Chapter 2 our axiomatic definition of a ’multi-normed space’ ((Eⁿ,||·||ₙ): n ∈ ℕ), where (E,||·||) is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum, maximum, and (p,q)-multi-norms based on a given space. Multi-norms measure ’geometrical features’ of normed spaces, in particular by considering their ’rate of growth’. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to ’multi-topological linear spaces’ through ’multi-null sequences’, and to ’multi-bounded’ linear operators, which are exactly the ’multi-continuous’ operators. We define a new Banach space ℳ(E,F) of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices. We conclude with a theory of ’orthogonal decompositions’ of a normed space with respect to a multi-norm, and apply this to construct a ’multi-dual’ space. Applications of this theory will be presented elsewhere.},
author = {H. G. Dales},
keywords = {Banach space; tensor products; Banach algebra; Banach lattice; $AL_p$-space; AM-space; positive operator; regular operator; Dedekind complete; Riesz space; Nakano property; multi-norm; multi-Banach space; dual multi-norm; maximum multi-norm; minimum multi-norm; matrices; tensor norms; condition (P); summing norms; weak p-summing norm; (p; q)-multi-norm; standard q-multi-norm; summing constant; multi-topological linear space; multi-null sequence; multi-bounded set; multi-bounded operator; multi-continuous operator; extensions of multi-norms; Hermitian decomposition; small decomposition; orthogonal decomposition; multi-dual space; multi-reflexive},
language = {eng},
title = {Multi-normed spaces},
url = {http://eudml.org/doc/285986},
year = {2012},
}

TY - BOOK
AU - H. G. Dales
TI - Multi-normed spaces
PY - 2012
AB - We modify the very well known theory of normed spaces (E,||·||) within functional analysis by considering a sequence (||·||ₙ: n ∈ ℕ) of norms, where ||·||ₙ is defined on the product space Eⁿ for each n ∈ ℕ. Our theory is analogous to, but distinct from, an existing theory of ’operator spaces’; it is designed to relate to general spaces $L^{p}$ for p ∈ [1,∞], and in particular to L¹-spaces, rather than to L²-spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory, that we shall use, we shall present in Chapter 2 our axiomatic definition of a ’multi-normed space’ ((Eⁿ,||·||ₙ): n ∈ ℕ), where (E,||·||) is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum, maximum, and (p,q)-multi-norms based on a given space. Multi-norms measure ’geometrical features’ of normed spaces, in particular by considering their ’rate of growth’. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to ’multi-topological linear spaces’ through ’multi-null sequences’, and to ’multi-bounded’ linear operators, which are exactly the ’multi-continuous’ operators. We define a new Banach space ℳ(E,F) of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices. We conclude with a theory of ’orthogonal decompositions’ of a normed space with respect to a multi-norm, and apply this to construct a ’multi-dual’ space. Applications of this theory will be presented elsewhere.
LA - eng
KW - Banach space; tensor products; Banach algebra; Banach lattice; $AL_p$-space; AM-space; positive operator; regular operator; Dedekind complete; Riesz space; Nakano property; multi-norm; multi-Banach space; dual multi-norm; maximum multi-norm; minimum multi-norm; matrices; tensor norms; condition (P); summing norms; weak p-summing norm; (p; q)-multi-norm; standard q-multi-norm; summing constant; multi-topological linear space; multi-null sequence; multi-bounded set; multi-bounded operator; multi-continuous operator; extensions of multi-norms; Hermitian decomposition; small decomposition; orthogonal decomposition; multi-dual space; multi-reflexive
UR - http://eudml.org/doc/285986
ER -