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:
“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)
“If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them.”
—Henry David Thoreau (1817–1862)
“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)