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:
 * $\norm f = 1$
 * $\map f x = \norm x$
 * $\map f y = 0$

Proof
Let $x \perp_B y$. Let $V' \subset V$ be the subspace spanned by $x$ and $y$. Define $ \overline{f}$ on $V'$ given by: $$ \overline{f} (a x + b y) = a \|x\|, ~ a, ~ b ~ \textit{scalars}$$ Clearly, $\overline{f}$ is linear and $\overline{f} (x) = \|x\|$, $\overline{f} (y) = 0$. Further: $$\|a x + b y\| = |a| \|x + \frac{b}{a} y\| \geq |a| \|x\| = |\overline{f} (a x + b y)|$$ 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 $\|f\| = \|\overline{f}\| = 1$ and thereby proving the necessity. Again if such a functional $f$ on $V$ exists, for any scalar $\lambda$: $$\| x + \lambda y\| \geq |f (x + \lambda y)| = \|x\|$$ establishing the sufficiency.