Separation of a Point and a Subspace - Imported Knowlpedia Content

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

Let $X$ be a normed space, let $Y\subset X$ be a subspace, and let $x_0\in X$ satisfy

d(x_0,Y):=\inf_{y\in Y}\|x_0-y\|=d>0.

Theorem (separating a point and a subspace): There exists a bounded linear functional $f:X\to\mathbb{K}$ such that

$f(y)=0$ for all $y\in Y$ ,
$\|f\|=1/d$ , and
$f(x_0)=1$ .

Context: This is a geometric consequence of Hahn–Banach in normed spaces. It produces a continuous hyperplane through $Y$ that separates $x_0$ from $Y$ .

Proof sketch (idea): Define a linear functional on $Y\oplus \operatorname{span}\{x_0\}$ (see direct sum) by mapping $y+\lambda x_0\mapsto \lambda$ and bound its norm using the distance assumption; then extend it by Hahn–Banach.