Statement
A function is said to be of class if it is k-times continuously differentiable. For example, class consists of continuous functions, and class consists of infinitely smooth functions.
Let f be of class on the interval . Assume furthermore that
for every function h that is of class on with h(a) = h(b) = 0. Then the fundamental lemma of the calculus of variations states that is identically zero on .
In other words, the test functions h ( functions vanishing at the endpoints) separate functions: is a Hausdorff space in the weak topology of pairing against functions that vanish at the endpoints.
Read more about this topic: Fundamental Lemma Of Calculus Of Variations
Famous quotes containing the word statement:
“He that writes to himself writes to an eternal public. That statement only is fit to be made public, which you have come at in attempting to satisfy your own curiosity.”
—Ralph Waldo Emerson (1803–1882)
“Most personal correspondence of today consists of letters the first half of which are given over to an indexed statement of why the writer hasn’t written before, followed by one paragraph of small talk, with the remainder devoted to reasons why it is imperative that the letter be brought to a close.”
—Robert Benchley (1889–1945)
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)