# Definition:Orthogonal (Hilbert Space)

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*This page is about Orthogonal in the context of Hilbert Space. For other uses, see Orthogonal.*

## Definition

Let $\HH$ be a Hilbert space.

Let $f, g \in \HH$.

Let $\innerprod f g = 0$, where $\innerprod \cdot \cdot$ denotes the inner product.

Then $f$ and $g$ are defined as being **orthogonal**:

- $f \perp g$

### Sets

Let $A, B \subseteq \HH$.

Then $A$ and $B$ are defined as **orthogonal** if and only if:

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

That is, if $a$ and $b$ are orthogonal elements of $A$ and $B$ for all $a \in A$ and $b \in B$.

This is denoted by $A \perp B$.

## Also known as

Two objects that are **orthogonal** are often seen described as **perpendicular**.

However, this is usually seen in the context of geometry, where those objects are straight lines.

## Sources

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