Norm satisfying Parallelogram Law induced by Inner Product/Corollary

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Corollary

Let $\struct {V, \norm \cdot}$ be a normed vector space over $\R$.


Then there exists an inner product $\innerprod \cdot \cdot : V \times V \to \R$ such that:

$\norm x = \sqrt {\innerprod x x}$

if and only if:

$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$

for each $x, y \in V$.


That is, a norm is induced by an inner product if and only if it satisfies the Parallelogram Law for Inner Product Spaces.


Proof

Necessary Condition

Suppose that there exists an inner product $\innerprod \cdot \cdot : V \times V \to \R$ such that:

$\norm x = \sqrt {\innerprod x x}$

Then $\struct {V, \innerprod \cdot \cdot}$ is an inner product space with inner product norm $\norm \cdot$.

So from the Parallelogram Law for Inner Product Spaces, we have:

$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$

for each $x, y \in V$.

$\Box$

Sufficient Condition

Suppose that:

$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$

Then, from Norm satisfying Parallelogram Law induced by Inner Product, there exists an inner product $\innerprod \cdot \cdot : V \times V \to \R$ such that:

$\norm x = \sqrt {\innerprod x x}$

$\blacksquare$


Sources