Functional analysis Hilbert spaces Proposition

Parallelogram law

Builds on Norm induced by an inner product

Statement

Let $V$ be an inner product space over $\R$ or $\C$ , with the induced norm $\norm{\cdot}$ . Then, for all $x, y \in V$ , $\norm{x + y}^2 + \norm{x - y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2.$

Remarks

The proof expands both squares with the inner product; the mixed terms cancel. By the Jordan–von Neumann theorem, a norm comes from an inner product if and only if it satisfies this law. The norm $\abs{x_1} + \abs{x_2}$ on $\R^2$ fails it for $x = (1, 0)$ and $y = (0, 1)$ , so it comes from no inner product.

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Vector space definition
Inner product definition
Norm definition
Inner product space definition
Norm induced by an inner product definition
Parallelogram law statement

Where this leads

9 pages build on this one.

Tag	`026HPGYGS4`
URI	`m2://026HPGYGS4`
Kind	statement, proposition
Writing	W1 stated
MSC	46C05
Status	This copy of the site has no build data. Run `m2 status 026HPGYGS4`.
Uses	Norm induced by an inner product
Same as	Wikidata Q919125 Wikipedia Parallelogram law

Parallelogram law

Statement

Remarks

Reading path

Where this leads

Further reading

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