# The Pythagorean Theorem for Inner Product Spaces Examples 1

Recall from The Pythagorean Theorem for Inner Product Spaces page that if $V$ is an inner product space and if $u, v \in V$ are such that $u$ and $v$ are orthogonal to each other, that is, $<u, v> = 0$, then:

(1)We will now look at some examples regarding the Pythagorean theorem for inner product spaces.

## Example 1

**Let $V$ be an inner product space. Let $u, v \in V$. Prove that $u$ and $v$ are orthogonal if and only if $\| u \| ≤ \| u + c v \|$ for every $c \in \mathbb{F}$.**

$\Rightarrow$ Suppose that $u$ and $v$ are orthogonal to each other. Then $<u, v> = 0$. Therefore by the Pythagorean theorem we have that:

(2)Clearly we have that $\| u \|^2 ≤ \| u \|^2 + \| cv \|^2 = \| u + cv \|^2$, and in squaring both sides we get that $\| u \| ≤ \| u + cv \|$.

$\Leftarrow$ Suppose that $\| u \| ≤ \| u + cv \|$ for every $c \in \mathbb{F}$. If we square both sides of this inequality then we have that:

(3)Now let $c = -b<u, v>$ where $b > 0$. Then we have that:

(4)If $v = 0$ then then we automatically have that $<u, v> = 0$. If $v \neq 0$, then let $b = \frac{1}{\| v \|^2}$ to get that:

(5)Therefore $\mid <u, v> \mid^2 = 0$ so $<u, v> = 0$.