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

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!


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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.