I have a doubt about the correctness of the following integral inequality. $\Vert \int_{a}^{b} f(x)g(x) dx\Vert\leq \Vert f(x)\Vert \Vert \int_{a}^{b}g(x) dx\Vert.$ Can any one justify my doubt?
question related to the modulus of a definite integral
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$\begingroup$
norm
integral-inequality
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1What is $x$ in $\Vert f(x)\Vert$ on the RHS? – 2017-01-18
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0Where did this come from? If anything, it should be $\|\int_{a}^{b} f(x)g(x) dx\|\leq \sup_{x\in[a,b]}\left(\| f(x)\|\right) \| \int_{a}^{b}g(x) dx\|$. But it's much more likely $\|\int_{a}^{b} f(x)g(x) dx\|\leq\sup_{x\in[a,b]}\left(\| f(x)\| \right)\int_{a}^{b}\|g(x)\| dx$ – 2017-01-18
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0yes..norm is over the interval of integration. Then the inequality is correct. – 2017-01-18