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:
“and the oxen near
The worn foundations of their resting-place,
The holy manger where their bed is corn
And holly torn for Christmas. If they die,
As Jesus, in the harness, who will mourn?
Lamb of the shepherds, Child, how still you lie.”
—Robert Lowell (1917–1977)
“I have seen how the foundations of the world are laid, and I have not the least doubt that it will stand a good while.”
—Henry David Thoreau (1817–1862)
“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o’ th’ day
The clock doth strike, by algebra.”
—Samuel Butler (1612–1680)