Norm satisfying Parallelogram Law induced by Inner Product/Corollary

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

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


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




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

for each $x, y \in V$.

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

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 $x, y \in V$.

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:


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