MVT - Math 508, Fall 2008 Jerry Kazdan Two Inequalities for...

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Unformatted text preview: Math 508, Fall 2008 Jerry Kazdan Two Inequalities for Integrals of Vector Valued Functions Theorem Let F : [ a , b ] → R n be a continuous vector-valued function. Then vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle Z b a F ( t ) dt vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ≤ Z b a bardbl F ( t ) bardbl dt with equality if and only if there is a continuous scalar valued function ϕ ( t ) ≥ 0 such that F ( t ) = ϕ ( t ) V where V : = R b a F ( t ) dt . Proof: We begin with the observation that for any vectors X and V negationslash = 0, the proof of the Schwarz inequality shows that ( X , V ) ≤ bardbl X bardblbardbl V bardbl with equality if and only if X = cV for some constant c ≥ 0. Thus if V is a constant vector, then for any t ( F ( t ) , V ) ≤ bardbl F ( t ) bardblbardbl V bardbl with equality if and only if F ( t ) = ϕ ( t ) V for some scalar valued function ϕ ( t ) ≥ 0....
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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