Definition:Birkhoff-James Orthogonality

Definition

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed linear space.

Let $x, y \in V$.

Then $x$ and $y$ are Birkhoff-James orthogonal if and only if

$\norm {x + \lambda y} \ge \norm y$

for all scalars $\lambda$.

This is denoted:

$x \perp_B y$

Source of Name

This entry was named for Garrett Birkhoff and Robert Clarke James.

Sources

• 1935: Garrett Birkhoff: Orthogonality in linear metric spaces (Duke Math. J. Vol. 1: pp. 169 – 172)
• 1947: Robert C. James: Orthogonality and linear functionals in normed linear spaces (Trans. Amer. Math. Soc. Vol. 61: pp. 265 – 292)