Characteristics of Birkhoff-James Orthogonality

Definition
Given a normed linear space $\mathbb{X}$ and $x,y\in\mathbb{X}$, $x$ is defined to be Birkhoff-James orthogonal to $y$, denoted by $x\perp_By$ if: $$\|x+\lambda y\|\geq \|y\|,~\textit{for every scalar}~\lambda$$

Theorem
Let $\mathbb{X}$ be a normed linear space and $x,y\in\mathbb{X}$. Then $x\perp_By$ if and only if $x=0$ or there is a continuous functional $ f$ on $\mathbb{X}$ such that $\| f\| = 1$, $\map f {x} = \|x\|$ and $\map f {y} = 0$.

Proof
Let $x \perp_B y$. Let $\mathbb{Y} \subset \mathbb{X}$ be the subspace spanned by $x$ and $y$. Define $ \overline{f}$ on $\mathbb{Y}$ 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 $\mathbb{X}$ such that $\|f\| = \|\overline{f}\| = 1$ and thereby proving the necessity. Again if such a functional $f$ on $\mathbb{X}$ exists, for any scalar $\lambda$: $$\| x + \lambda y\| \geq |f (x + \lambda y)| = \|x\|$$ establishing the sufficiency.