Orthogonal and Orthonormal Sets in Inner Product Spaces

Orthogonal and Orthonormal Sets in Inner Product Spaces

Definition: Let $H$ be an inner product space and let $S \subseteq H$. Then $S$ is said to be an Orthogonal Set if: a) $\langle x, y \rangle = 0$ for all $x, y \in S$ with $x \neq y$.

Definition: Let $H$ be an inner product space and let $S \subseteq H$. Then $S$ is said to be Orthonormal Set if both: a) $\langle x, y \rangle = 0$ for all $x, y \in S$ with $x \neq y$. b) $\| x \| = 1$ for all $x \in S$.

Of course, $\| \cdot \|$ is the norm induced by the inner product, that is, $\| x \| = \langle x, x \rangle^{1/2}$ for every $x \in X$.