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$\int_{b}^{a}p(x)(u'(x)v'(x))+q(x)(u(x)v(x))dx \leq [\int_{b}^{a}p(x)(u'(x)u'(x))+q(x)(u(x)u(x))dx]^{\frac{1}{2}}[\int_{b}^{a}p(x)(v'(x)v'(x))+q(x)(v(x)v(x))dx]^{\frac{1}{2}}$

I have to show something is a norm, and the above is the last step, but I have no idea how to prove this, does anyone can help me? Thanks!

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This is just an application of the (slightly generalized) Cauchy-Schwarz inequality.

Let $(\cdot,\cdot) : X \times X \to \mathbb{R}$ be a bilinear form, such that $(x,x) \ge 0$ for all $x \in X$. Then, $$(x,y) \le (x,x)^{1/2} \, (y,y)^{1/2}.$$

The proof is almost the same as for an definite inner product (e.g. with $(x,x) > 0$ for all $x \in X \setminus \{0\}$). It can be found in the German wikipedia (it is not contained in the English wikipedia, but nevertheless it should be understandable).

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