A natural proof of the Cauchy-Schwarz inequality

Solution 1:

There is also an approach by "amplification" which is really cool. Also the exact same trick works to prove Hölder's inequality and is generally a very important principle for improving inequalities.

It goes like this: We start out with $$\langle a-b,a-b\rangle\ge 0$$ for $a,b$ in your inner product space, and $a\not=0$, $b\not=0$. This implies $$2\langle a,b\rangle\le \langle a, a\rangle + \langle b, b\rangle$$ Now notice that the left hand side is invariant under the scaling $a\mapsto \lambda a$, $b\mapsto \lambda^{-1}b$ for $\lambda>0$. This gives $$2\langle a,b\rangle \le \lambda^2 \langle a,a\rangle + \lambda^{-2}\langle b, b\rangle$$ Now look at the right hand side as a function of the real variable $\lambda$ and find the optimal value for $\lambda$ using calculus (set the derivative to $0$):

$$\lambda^2=\sqrt{\frac{\langle b,b\rangle}{\langle a,a\rangle}}$$

Plugging this value in, we obtain

$$2\langle a,b\rangle\le \sqrt{\langle a,a\rangle}\sqrt{\langle b,b\rangle}+\sqrt{\langle a,a\rangle}\sqrt{\langle b,b\rangle}$$

i.e.

$$\langle a,b\rangle\le\sqrt{\langle a,a\rangle}\sqrt{\langle b,b\rangle}$$

Notice how we took a trivial observation and "optimized" the expression by exploiting scaling invariance.

Solution 2:

Recall the Pythagorean theorem: If $u_1, \cdots u_n$ are pairwise orthogonal, then $$ \| u_1 + \cdots u_n \|^2 = \|u_1\|^2 + \cdots + \| u_n \|^2.$$

I want to use this to tell us something about two non-zero vectors $u$ and $v,$ but they aren't necessarily orthogonal. So consider the projection of $u$ onto the plane of vectors orthogonal to $v:$ $$w = u - \frac{ \langle u,v \rangle}{\|v\|^2} v.$$ This is certainly orthogonal to $v,$ and the Pythagorean theorem applied to $w$ and $v$ gives the Cauchy-Schwarz inequality.

Solution 3:

The inequality $| \langle a, b \rangle | \leq \| a \| \| b \|$ can be rewritten as \begin{equation} | \langle a, \frac{b}{\|b\|} \rangle | \leq \| a \| \end{equation} (assuming $b \neq 0$).

On the left we see the component of $a$ on the unit vector $u = \frac{b}{\| b \|}$. On the right we have the norm of $a$. Of course the norm of $a$ is larger than the component of $a$ on $u$, this is very intuitive. To make this into a rigorous proof, we merely have to write $a = \langle a, u \rangle u + v$, note that $v \perp u$, and use the pythagorean theorem.

Another approach is to start with the inequality \begin{equation} 0 \leq \| a - \text{proj}_b(a) \|^2. \end{equation}

Just expand the right hand side and Cauchy-Schwarz pops out.

Solution 4:

This is not exactly an answer but an explanation for the idea behind the Ragib Zaman's answer.

Let $K$ be the field of real numbers or the field of complex numbers. Let $E$ be a pre-Hilbert space over $K$.

Let $x, y$ be elements of $E$ such that $\langle x, y \rangle = 0$. Then $\|x + y\|^2 = \langle x+y, x+ y\rangle = \langle x, x\rangle + \langle x, y \rangle + \langle y, x\rangle + \langle y,y\rangle = \|x\|^2 + \|y\|^2$.

Let $x, y$ be elements of $E$ such that $\langle x - y, y \rangle = 0$. Then, by the above formula, $\|x\|^2 = \|x - y\|^2 + \|y\|^2$. Hence $\|x\| \ge \|y\|$.

Finally let $u, v$ be non-zero elements of $E$. Let $t$ be an element of $K$ such that $\langle u - tv, v \rangle = 0$. $t$ must be $\frac{ \langle u,v \rangle}{\langle v, v\rangle} $ since $v \neq 0$. Then $\|u\| \ge \|tv\|$ by the previous inequality. This leads to the Cauchy-Schwarz inequality immediately.

Solution 5:

Here's my favorite proof, mainly because it's nicely symmetric, easy to remember and not impossible to come up with (the main trick is that $2=1+1$):

We want to prove $$|\langle x,y\rangle|\le\|x\|\|y\|$$ This is linear in $x$ or $y$ and obviously holds for $x=0$ or $y=0$. Therefore without loss of generality we can suppose that $\|x\|=\|y\|=1$: $$|\langle x,y\rangle|\le\|x\|\|y\|\Longleftrightarrow|\langle x,y\rangle|\le1\Longleftrightarrow2|\langle x,y\rangle|\le2\Longleftrightarrow2|\langle x,y\rangle| \le \|x\|^2+\|y\|^2$$

Let $u\in\mathbb C$ be a complex unit (i.e. $|u|=1$), then $$\begin{align}0\le\|x-uy\|^2&=\langle x-uy,x-uy\rangle=\langle x,x\rangle-u\langle x,y\rangle-\overline{u\langle x,y\rangle}+\langle y,y\rangle\\&=\|x\|^2+\|y\|^2-2\,\text{Re}(u\langle x,y\rangle)\end{align}$$ Now just set $u$ so that $\text{Re}(u\langle x,y\rangle)=|\langle x,y\rangle|$ and you are done.

Of course in the real case you can just expand $0\le\|x\pm y\|^2$.