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