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Beta Dual
In functional analysis and related areas of mathematics, the beta-dual or -dual is a certain linear subspace of the algebraic dual of a sequence space. Definition Given a sequence space the -dual of is defined as :X^:= \left \. If is an FK-space then each in defines a continuous linear form on :f_y(x) := \sum_^ x_i y_i \qquad x \in X. Examples * c_0^\beta = \ell^1 * (\ell^1)^\beta = \ell^\infty * \omega^\beta = \ Properties The beta-dual of an FK-space is a linear subspace of the continuous dual of . If is an FK-AK space In functional analysis and related areas of mathematics an FK-AK space or FK-space with the AK property is an FK-space which contains the space of finite sequences and has a Schauder basis. Examples and non-examples * c_0 the space of converge ... then the beta dual is linear isomorphic to the continuous dual. {{mathanalysis-stub Functional analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |