Polarization Identity/Real Vector Space
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Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\R$.
Let $\norm \cdot$ be the inner product norm for $V$.
Then we have:
- $4 \innerprod x y = \norm {x + y}^2 - \norm {x - y}^2$
for all $x, y \in V$.
Proof
We have:
\(\ds \norm {x + y}^2 - \norm {x - y}^2\) | \(=\) | \(\ds \innerprod {x + y} {x + y} - \innerprod {x - y} {x - y}\) | Definition of Inner Product Norm | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\innerprod x {x + y} + \innerprod y {x + y} } - \paren {\innerprod x {x - y} - \innerprod y {x - y} }\) | since an inner product is linear in the first argument | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\innerprod {x + y} x + \innerprod {x + y} y} - \paren {\innerprod {x - y} x - \innerprod {x - y} y}\) | since a real inner product is symmetric | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\innerprod x x + \innerprod y x + \innerprod x y + \innerprod y y} - \paren {\innerprod x x - \innerprod y x - \innerprod x y + \innerprod y y}\) | using linearity in the first argument | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \innerprod x y + 2 \innerprod y x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \innerprod x y\) | since a real inner product is symmetric |
$\blacksquare$
Sources
- 1965: Michael Spivak: Calculus on Manifolds ... (previous) ... (next): 1. Functions on Euclidean Space: Norm and Inner Product
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $8.3$: Properties of the Induced Norms