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