5
$\begingroup$

I am working through linear algebra problems in Apostol's Calculus, and he has numerous problems that seem to imply that Cauchy-Schwarz holds no matter how the inner product is defined. Then, he has problems where the triangle inequality holds despite alternative definitions of vector norm. This got me thinking, since the proof of the triangle inequality in Apostol relies on Cauchy-Schwarz, that the triangle inequality would hold regardless of how the vector norm is defined (if it involves the dot product).

I then found this response to a question, which states what I was thinking.

Are there any proofs that the Cauchy-Schwarz inequality holds in any inner product space (I looked for some and found none and could not prove it myself)? I've had a semester of algebra (Artin) and some analysis, if there is a proof at such a level of understanding. Intuitive explanations are good, too.

$\endgroup$
  • 3
    $\begingroup$ @AWertheim: That link seems like a perfectly reasonable thing to post as an answer ;) $\endgroup$ – Jim Aug 8 '13 at 20:42
  • 1
    $\begingroup$ @Jim done, and with the advantage of nicer formatting! Cheers :) $\endgroup$ – Alex Wertheim Aug 8 '13 at 20:44
  • $\begingroup$ "Why does it hold in any space?" could reasonably be taken to mean "Why is there any space in which it holds?". But I doubt that's what you meant. Just changing "any" to "every" would make it completely unambiguous. $\endgroup$ – Michael Hardy May 23 '14 at 19:14
  • $\begingroup$ This post is an exceptional proof with good exposition. I probably wouldn't have come up with it myself any faster than the usual discriminant proof but it feels a lot more natural after the fact. Also, a nice formalism for expressing the intuitive idea of "in the worst case". $\endgroup$ – Eric Stucky May 23 '14 at 19:29
6
$\begingroup$

Wikipedia has the one I've usually seen.

Cauchy-Schwarz Inequality

Nothing especially tricky about it, just a really clever setup.

$\endgroup$
3
$\begingroup$

The proof on Wikipedia is valid, and is instructive because it uses ideas which are applicable elsewhere in linear algebra. The proof I've usually seen selects vectors u,v and defines

$$f(t) = \langle u + t v , u + t v \rangle$$

and notices that f is nonnegative for all t, by the positive definiteness of the inner product. Expanding out the inner product by bilinearity and combining like terms by symmetry, you get a quadratic function of t:

$$f(t) = \| v \|^2 t^2 + 2 \langle u,v\rangle t + \| u \|^2.$$

(In the complex case you have to do something slightly different.)

This is nonnegative, so its discriminant is nonpositive, in other words:

$$4 \langle u,v\rangle^2 - 4 \| u \|^2 \| v \|^2 \leq 0$$

from which the inequality follows.

An equivalent way of looking at this (for $v \neq 0$) is to find the minimum of $f$. This occurs at $t=-\langle u , v \rangle/\| v \|^2$. So:

$$\langle u , v \rangle^2/\| v \|^2 - 2 \langle u , v \rangle^2/\| v \|^2 + \| u \|^2 = \| u \|^2 - \langle u , v \rangle^2/\| v \|^2 \geq 0$$

A simple rearrangement yields the Cauchy-Schwarz inequality. This is somewhat like the proof on Wikipedia, because $u-\langle u,v \rangle/\| v \|^2 v$ is the projection of $u$ onto the orthogonal complement of the span of $v$.

$\endgroup$
  • 2
    $\begingroup$ Ian, I think the spacing looks better this way. If you disagree, please let me know, and I will roll it back. $\endgroup$ – Jyrki Lahtonen May 23 '14 at 19:22
2
$\begingroup$

This is an old question but here is an interesting proof that generalizes it a bit. This is adapted from Folland's Real Analysis. Consider some unit scalar $\alpha$ (that is $|\alpha| = 1$) and some $t \in (0,\infty)$. Then,

$$\begin{align*} \langle x -\alpha ty, x-\alpha ty\rangle &= \langle x, x-\alpha ty\rangle + \langle -\alpha ty, x-\alpha ty\rangle \\ &= \langle x,x\rangle - \left(\overline{t} \langle x, \alpha y\rangle +t \langle \alpha y,x\rangle\right) + \underbrace{\alpha\overline{\alpha}}_{=|\alpha|^2=1} t^2 \langle y,y\rangle \\ &= \langle x,x\rangle - \underbrace{\left(\overline{t \langle x, \alpha y\rangle} + t \langle x,\alpha y\rangle\right)}_{2t\operatorname{Re} \langle x,y\rangle} + t^2 \langle y,y\rangle \\ &= \|y\|^2 t^2 - 2\operatorname{Re} \langle x,\alpha y\rangle t + \|x\|^2. \end{align*}$$

Now, this is a quadratic in $t$, and we observe that it opens upward and is non-negative. Therefore, consider its discriminant, $b^2 -4ac \le 0$.

We have $b = 2\operatorname{Re} \langle x,\alpha y\rangle$. However, since $\alpha = \frac{z}{\|z\|} = \operatorname{sgn} z$ for some $z$, we have $\langle x, \alpha y\rangle = \langle \alpha y, x\rangle = |\langle x, y\rangle|$; hence $\operatorname{Re}\langle x,\alpha y\rangle = |\langle x, y\rangle|$.

Therefore, we obtain

$$4|\langle x,y\rangle|^2 \le 4\|x\|^2\|y\|^2 \implies |\langle x,y\rangle| \le \|x\|\|y\|.$$

$\endgroup$
  • $\begingroup$ I enjoy this because there is something fun about a proof for a concept that generalizes to fairly advanced structures using techniques developed in middle-school. $\endgroup$ – Emily May 3 '15 at 15:45
  • $\begingroup$ how does $\langle x, \alpha y\rangle=\langle\alpha y, x\rangle=|\langle x, y\rangle|$ hold? $\endgroup$ – user149418 Feb 15 '18 at 16:28
0
$\begingroup$

Let $t=\operatorname{sign}(\langle v,w\rangle)$, then $t\in \{-1,0,1\}$ and by triangle inequality, $|v+tw|\leq |v|+|t||w|\leq |v|+|w|$, thus $$|v + tw|^2=|v|^2+2t\langle v,w\rangle+|w|^2\leq |v|^2+2|v||w|+|w|^2.$$ Hence $$t\langle v,w\rangle=|\langle v,w\rangle|\leq |v||w|.$$

$\endgroup$
  • 1
    $\begingroup$ Try proving the triangle inequality without Cauchy-Schwarz. $\endgroup$ – Unit Dec 29 '14 at 18:32

Your Answer

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