Definition:Orthogonal (Linear Algebra)/Sets

Definition

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $A, B \subseteq V$.

We say that $A$ and $B$ are 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 $b \in B$.

We write:

$A \perp B$

Also denoted as

In the case where $A = \set a$ is a singleton, the notation $a \perp B$ is often encountered.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Definition $2.1$