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.

Status in mathematical logic

Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basis for such spaces may require Zorn's lemma. Many very important theorems require the Hahn-Banach theorem which itself requires Zorn's lemma in case of a general infinite-dimensional space.

Related Topics:
Vector space basis - Zorn's lemma - Hahn-Banach theorem

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