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
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