Exercise 6.11

Question

Exercise 11: Let $\alpha$ be a fixed increasing function on $[a,b]$. For $u\in\mathscr R(\alpha)$, define $$||u||_2=\biggl(\int_a^b|u|^2\,d\alpha\biggr)^{1/2}.$$ Suppose
    $f,g,h\in\mathscr R(\alpha)$, and prove the triangle inequality $$||f-h||_2\le||f-g||_2+||g-h||_2$$ as a consequence of the Schwarz inequality, as in the proof of Theorem 1.37.

analambanomenos · Accepted Answer

As in the proof of Theorem 1.37(f), we show that
    \begin{align*}
        ||u+v||_2^2 &=\int_a^b|u+v|^2\,d\alpha \
        &\le \int_a^b\bigl(|u|^2+2|uv|+|v|^2\bigr)\,d\alpha \
        &= ||u||_2^2+2\int_a^b|uv|\,d\alpha+||v||_2^2 \
        &\le ||u||_2^2+2||u||_2||v||_2+||v||_2^2 \
        &= \bigl(||u||_2+||v||_2\bigr)^2
    \end{align*}
    so that $||u+v||_2\le||u||_2+||v||_2$. The result follows by letting $u=f-g$ and $v=g-h$.

Exercise 6.11

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Exercise 6.11

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