4
$\begingroup$

It is well known that if $X$ is a Hilbert space and $W \subset X$ is a closed subspace, then for each $x \notin W$, there is a unique element of $W$ which lies at minimal distance from $x$. However, I could not think of an explicit example to show that this fails for a Banach space which is not a Hilbert space. Does anyone have one?

2 Answers 2

3

Pretty much the simplest nontrivial example works. Take $X = \mathbb{R}^2 = \text{span}(e_1, e_2)$ with the sup norm, $W = \text{span}(e_1)$, and $x = e_2$. The minimal distance is $1$ and it is achieved by all elements of the form $a e_1, a \in [-1, 1]$.

4

Banach spaces having this property are called proximinal. This property holds for any strictly convex reflexive Banach Space (such as the $L^p$ spaces).

The existence of the minimum owes to a compactness result known as the Alaoglu theorem. Strict convexity will yield uniqueness.