A convex functional is a mapping from a vector space to the real numbers that satisfies the property of convexity, meaning that for any two points in its domain, the functional's value at any point on the line segment connecting these points is less than or equal to the weighted average of its values at those two points. This concept plays a crucial role in variational principles and calculus of variations, as convex functionals often lead to well-defined optimization problems and solutions, helping identify extremal functions that minimize or maximize the functional's value.
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