Functional analysis

Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. It has its historical roots in the study of transformations such as the Fourier transform and in the study of differential and integral equations. The word 'functional' goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to Volterra.

Banach spaces

General Banach spaces are more complicated.

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There is no clear definition of what would constitute a base, for example.

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For any real number p ≥ 1, an example of a Banach space is given by "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral" (see Lp spaces).

Related Topics:
Lebesgue-measurable function - Absolute value - L^''p'' spaces

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In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphism from a space into its dual's dual. This is explained in the dual space article.

Related Topics:
Dual space - Continuous

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The notion of derivative is extended to arbitrary functions between Banach spaces; it turns out that the derivative of a function at a certain point is really a continuous linear map.

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