Category:Norm satisfying Parallelogram Law induced by Inner Product

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This category contains pages concerning Norm satisfying Parallelogram Law induced by Inner Product:


Let $V$ be a vector space over $\R$.

Let $\norm \cdot : V \to \R$ be a norm on $V$ such that:

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

for each $x, y \in V$.


Then the function $\innerprod \cdot \cdot : V \times V \to \R$ defined by:

$\ds \innerprod x y = \frac {\norm {x + y}^2 - \norm {x - y}^2} 4$

for each $x, y \in V$, is an inner product on $V$.

Further, $\norm \cdot$ is the inner product norm of $\struct {V, \innerprod \cdot \cdot}$.

Pages in category "Norm satisfying Parallelogram Law induced by Inner Product"

The following 3 pages are in this category, out of 3 total.