Orthogonal Sets in an Inner Product Space

# Orthogonal Sets in an Inner Product Space

 Definition: Let $H$ be an inner product space. Two elements $x, y \in H$ are said to be Orthogonal denoted $x \perp y$ if $\langle x, y \rangle = 0$.
 Definition: Let $H$ be an inner product space, $x \in H$, and $S \subseteq H$. Then $x$ is said to be Orthogonal to $S$ denoted $x \perp S$ if $\langle x, y \rangle = 0$ for every $y \in S$.

For example, consider the space $\mathbb{R}^2$ and the vector $(1, 1) \in \mathbb{R}^2$. Let $(x, y) \in \mathbb{R}^2$ and consider the inner product:

(1)
\begin{align} \quad \langle (1, 1), (x, y) \rangle = 1 \cdot x + 1 \cdot y = x + y \end{align}

Then $(1, -1)$ orthogonal to $(1, 1)$, that is:

(2)
\begin{align} \quad (1, 1) \perp (1, -1) \end{align}

Furthermore, if $S = \{ (x, y) \in \mathbb{R}^2 : x + y = 0 \}$, then:

(3)
\begin{align} \quad (1, 1) \perp S \end{align}
 Definition: Let $H$ be an inner product space and $S \subseteq H$. Then the Orthogonal Set of $S$ denoted $S^{\perp}$ is the set of all vectors $x \in H$ that are orthogonal to $S$, that is, $S^{\perp} = \{ x \in H : x \perp S \}$.