The Normed Space Induced by an Inner Product

The Normed Space Induced by an Inner Product

Theorem 1: Let $H$ be an inner produce space. Then the function $\| \cdot \| : H \to \mathbb{R}$ defined for all $x \in H$ by $\| x \| = \langle x, x \rangle^{1/2}$ is a norm on $H$.
  • Proof: We show that $\| \cdot \|$ has all of the properties of a norm.
  • First, suppose that $x = 0$. Then $\| 0 \| = \langle 0, 0 \rangle^{1/2} = 0$. Now suppose that $\| x \| = 0$. Then $\langle x, x \rangle = 0$. But by definition, this means that $x = 0$. Hence $\| x \| = 0$ if and only if $x = 0$.
  • Second, let $x \in H$ and let $\lambda \in \mathbb{C}$. Then:
(1)
\begin{align} \quad \| \lambda x \| = \langle \lambda x, \lambda x \rangle^{1/2} = \left ( \lambda \overline{\lambda} \langle x, x \rangle \right )^{1/2} = (|\lambda|^2 \langle x, x \rangle )^{1/2} = |\lambda| \langle x, x \rangle^{1/2} = | \lambda | \| x \| \end{align}
  • Lastly, let $x, y \in H$. Then by the Cauchy-Schwarz inequality we have that:
(2)
\begin{align} \quad \| 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 \\ &= \langle x, x \rangle + \langle x, y \rangle + \overline{\langle x, y \rangle} + \langle y, y \rangle \\ &= \langle x, x \rangle + 2 \mathrm{Re} \langle x, y \rangle + \langle y, y \rangle \\ & \leq \| x \|^2 + 2 |\langle x, y \rangle| + \| y \|^2 \\ & \leq \| x \|^2 + 2 \langle x, x \rangle^{1/2} \langle y, y \rangle^{1/2} + \| y \|^2 \\ & \leq \| x \|^2 + 2 \| x \| \| y \| + \| y \|^2 \\ & \leq (\| x \| + \| y \|)^2 \end{align}
  • Therefore:
(3)
\begin{align} \quad \| x + y \| \leq \| x \| + \| y \| \end{align}
  • Therefore $\| \cdot \| : H \to \mathbb{R}$ is a norm. $\blacksquare$

So whenever we have an inner product space $H$ we can obtain a norm on $H$. We give this norm a special name defined below.

Definition: Let $H$ be an inner product space. The Norm Induced by the Inner Product on $H$ is the norm $\| \cdot \| : H \to \mathbb{R}$ defined for all $x \in H$ by $\| x \| = \langle x, x \rangle^{1/2}$.
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