Definition:Orthogonal (Hilbert Space)

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


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:

$f \perp g$


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.