Characteristics of Birkhoff-James Orthogonality

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

Let $x, y \in V$.

Then $x$ and $y$ are Birkhoff-James orthogonal either:


 * $(1): \quad x = 0$

or:
 * $(2): \quad$ there exists a continuous functional $ f$ on $\struct {V, \norm {\,\cdot\,} }$ such that:
 * $\norm f = 1$
 * $\map f x = \norm x$
 * $\map f y = 0$

Necessary Condition
Let $x \perp_B y$.

Let $V' \subset V$ be the subspace spanned by $x$ and $y$.

Define $\overline f$ on $V'$ as:
 * $\map {\overline f} {a x + b y} = a \norm x$

for $a$ and $b$ scalars.

Clearly, $\overline f$ is linear and:
 * $\map {\overline f} x = \norm x$
 * $\map {\overline f} y = 0$

Further:

proving that $\overline f$ is a bounded functional of norm $1$.

Now by Hahn-Banach Theorem, $\overline f$ can be extended to a functional $f$ on $V$ such that $\norm f = \norm {\overline f} = 1$

This proves the necessity.

Sufficient Condition
Let such a functional $f$ on $V$ exist.

Then for any scalar $\lambda$:
 * $\norm {x + \lambda y} \ge \size {\map f {x + \lambda y} } = \norm x$

establishing the sufficiency.