Foundations of Mathematics Considerations
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. However, a somewhat different concept, Schauder basis, is usually more relevant in functional analysis. Many very important theorems require the Hahn–Banach theorem, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.
Read more about this topic: Functional Analysis
Famous quotes containing the words foundations of and/or foundations:
“The world must be made safe for democracy. Its peace must be planted upon the tested foundations of political liberty.”
—Woodrow Wilson (1856–1924)
“The objects of a financier are, then, to secure an ample revenue; to impose it with judgment and equality; to employ it economically; and, when necessity obliges him to make use of credit, to secure its foundations in that instance, and for ever, by the clearness and candour of his proceedings, the exactness of his calculations, and the solidity of his funds.”
—Edmund Burke (1729–1797)