Parallelogram Law (Inner Product Space)

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Theorem

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $\norm \cdot$ be the inner product norm of $\struct {V, \innerprod \cdot \cdot}$.

Let $f, g \in V$ be arbitrary.


Then:

$\norm {f + g}^2 + \norm {f - g}^2 = 2 \paren {\norm f^2 + \norm g^2}$


Proof

\(\ds \norm {f + g}^2 + \norm {f - g}^2\) \(=\) \(\ds \innerprod {f + g} {f + g} + \innerprod {f - g} {f - g}\) Definition of Inner Product Norm
\(\ds \) \(=\) \(\ds \innerprod f f + \innerprod f g + \innerprod g f + \innerprod g g + \innerprod f f - \innerprod f g - \innerprod g f + \innerprod g g\) Linearity of Inner Product
\(\ds \) \(=\) \(\ds 2 \innerprod f f + 2 \innerprod g g\)
\(\ds \) \(=\) \(\ds 2 \paren {\norm f^2 + \norm g^2}\) Definition of Inner Product Norm

$\blacksquare$


Also see


Sources