Question: Prove that defines an inner product on the space C'[a, b] of continuously differentiable functions on the interval [a,b]. Write out the corresponding norm, known

Prove that

defines an inner product on the space C'[a, b] of continuously differentiable functions on the interval [a,b]. Write out the corresponding norm, known as the Sobolev H' norm; it and its generalizations play an extremely important role in advanced mathematical analysis, [36],

lf,g)= 1 [f(x)g(x)+/,(x) g,(x)] dx

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