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:
“Foundations of Morality: are like all other foundations; if you dig too much about them the superstructure will come tumbling down.”
—Samuel Butler (1835–1902)
“Neutrality is a negative word. It does not express what America ought to feel.... We are not trying to keep out of trouble; we are trying to preserve the foundations on which peace may be rebuilt.”
—Woodrow Wilson (1856–1924)
“It is a monstrous thing to force a child to learn Latin or Greek or mathematics on the ground that they are an indispensable gymnastic for the mental powers. It would be monstrous even if it were true.”
—George Bernard Shaw (1856–1950)