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, foundations and/or mathematics:
“Society is held together by our need; we bind it together with legend, myth, coercion, fearing that without it we will be hurled into that void, within which, like the earth before the Word was spoken, the foundations of society are hidden.”
—James Baldwin (1924–1987)
“The aggregate of all knowledge has not yet become culture in us. Rather it would seem as if, with the progressive scientific penetration and dissection of reality, the foundations of our thinking grow ever more precarious and unstable.”
—Johan Huizinga (1872–1945)
“... though mathematics may teach a man how to build a bridge, it is what the Scotch Universities call the humanities, that teach him to be civil and sweet-tempered.”
—Amelia E. Barr (1831–1919)