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Dual Norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Definition Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous dual space. The dual norm of a continuous linear functional f belonging to X^* is the non-negative real number defined by any of the following equivalent formulas: \begin \, f \, &= \sup &&\ \\ &= \sup &&\ \\ &= \inf &&\ \\ &= \sup &&\ \\ &= \sup &&\ \;\;\;\text X \neq \ \\ &= \sup &&\bigg\ \;\;\;\text X \neq \ \\ \end where \sup and \inf denote the supremum and infimum, respectively. The constant 0 map is the origin of the vector space X^* and it always has norm \, 0\, = 0. If X = \ then the only linear functional on X is the constant 0 map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal \sup \varnothing = - \infty instead of the correct value of 0. The ma ... [...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] |