Pythagorean theorem in an inner product space

Builds on Orthogonal vectors Norm induced by an inner product

Statement

Let $V$ be an inner product space over $\R$ or $\C$ , with the induced norm $\norm{\cdot}$ . If $x, y \in V$ are orthogonal, then $\norm{x + y}^2 = \norm{x}^2 + \norm{y}^2.$

Remarks

The proof: $\norm{x + y}^2 = \norm{x}^2 + 2 \operatorname{Re} \langle x, y \rangle + \norm{y}^2$ . For pairwise orthogonal $x_1, \dots, x_k$ , $\norm{x_1 + \cdots + x_k}^2 = \norm{x_1}^2 + \cdots + \norm{x_k}^2$ . In a real space, the converse holds; in a complex space, the equality only gives $\operatorname{Re} \langle x, y \rangle = 0$ .

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Vector space definition
Inner product definition
Norm definition
Inner product space definition
Orthogonal vectors definition
Norm induced by an inner product definition
Pythagorean theorem in an inner product space statement

Tag	`0BVTDM872V`
URI	`m2://0BVTDM872V`
Kind	statement, theorem
Writing	W0 stub
MSC	46C05
Status	This copy of the site has no build data. Run `m2 status 0BVTDM872V`.
Uses	Orthogonal vectors, Norm induced by an inner product

Pythagorean theorem in an inner product space

Statement

Remarks

Reading path

Further reading

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