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
- Norm satisfying Parallelogram Law induced by Inner Product establishes a converse result: if a norm satisfies the parallelogram law, then it is induced by an inner product.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): parallelogram law: 1.
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: $2.3$ Parallelogram Law
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $8.3$: Properties of the Induced Norms