# Definition:Orthogonal (Hilbert Space)

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## Definition

Let $H$ be a Hilbert space, and let $f, g \in H$.

If $\innerprod f g = 0$, say $f$ and $g$ are **orthogonal**; in symbols, write $f \perp g$.

One also frequently encounters the terminology that $f$ **is perpendicular to** $g$ for the same concept.

### Sets

Let $A, B \subseteq H$. Then say $A$ and $B$ are **orthogonal** if and only if:

- $\forall a \in A, b \in B: a \perp b$

This is denoted by $A \perp B$.

When $A, B$ are closed linear subspaces of $H$, they are called **orthogonal subspaces**.

If $A$ consists of only one element $a$, also $a \perp B$ is encountered.

## Orthogonal Complement

Let $A \subseteq H$. Then the set

- $A^\perp := \set {f \in H: f \perp A}$

is called the **orthogonal complement** or **orthocomplement** of $A$.

The operation of assigning $A^\perp$ to $A$ is referred to as **orthocomplementation**.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next): $I.2.1$