Norms and Inner products

Due by Fri, 12 Jan 2018

In this exercise, we explore further the relation between norms and inner-products. Let $V$ be a vector space over $\mathbb{R}$.

We shall see that:

• We can associate to each inner product $\langle\cdot, \cdot \rangle$ on $V$ a norm $\Vert\cdot\Vert$ on $V$.
• Such a norm satisfies some useful properties which we take as the definition of a norm on $V$.
• A norm derived from an inner-product satisfies an additional property called parallelogram law.
• Given the norm on $V$ associated to an inner-product, we can recover the inner-product using polarization.
• However, not all norms are associated to inner-products.

Exercises

1. Let $\langle\cdot, \cdot \rangle$ be an inner-product on $V$ and define $\Vert x \Vert = \sqrt{\langle x, x\rangle}$ for $x \in V$. Show that
   1. for all $x\in V$, $\Vert x\Vert \geq 0$ with equality if and only if $x = 0$,
   2. for all $x \in V$ and $c \in \mathbb{R}$, $\Vert c \cdot x\Vert = \vert c \vert \cdot \Vert x\Vert$,
   3. (triangle inequality) for all $x, y \in V$, $\Vert x + y\Vert \leq \Vert x\Vert + \Vert y \Vert$.

   $~$

   We take these properties to be the definition of a norm, i.e., a norm is a function $\Vert\cdot\Vert: V \to \mathbb{R}$ that satisfies the properties (1) - (3).

2. Let $\langle\cdot, \cdot \rangle$ be an inner-product on $V$ and define $\Vert x \Vert = \sqrt{\langle x, x\rangle}$ for $x \in V$. Show that for $x, y \in V$,

   $$\Vert x + y \Vert^2 + \Vert x - y \Vert^2 = 2(\Vert x\Vert^2 + \Vert y \Vert^2).$$

   This is called the parallelogram law.

3. Let $V = \mathbb{R^2}$ and define $\Vert (x, y)\Vert = \vert x \vert + \vert y \vert$. Show that $\Vert\cdot\Vert$ satisfies the properties (1) - (3) of problem 1 (i.e., it is a norm), but not the parallelogram law (and is thus not obtained from an inner product).

4. Suppose a norm $\Vert\cdot\Vert$ on a vector space $V$ is of the form $\Vert x \Vert = \sqrt{\langle x, x\rangle}$ for an inner product $\langle\cdot, \cdot \rangle$, then show that the inner product is given by

   $$\langle x, y \rangle = \frac{\Vert x + y \Vert^2 - \Vert x - y \Vert^2}{4}.$$

   This is called the polarization identity.

Remark: In fact, if a norm satisfies the parallelogram law, then the polarization identity gives an inner product (this is harder to prove than the above statements).

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