Definition:Orthogonal (Hilbert Space)
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 $A \perp B$.