Let $U \subset \mathbb{R}^n$ be a subspace, let $\lVert \cdot \rVert_A$ be any norm on $\mathbb{R}^n$, let $a \in \mathbb{R}^n$. Because of Bolzano–Weierstrass, there exists a $v \in U$ such that $$ \lVert v-a \rVert_A = \inf_{u\in U} \lVert u-a \rVert_A $$
- Is the choice of $v$ independent of the choice of the norm?
- If not, is $v \in U$ always unique with this property, no matter what norm is chosen?
(If the answer to 1 is yes, it follows that $v$ is unique because of the situation with the euclidian norm).
I feel like 1. is false, but I failed to construct any counterexample. If it is true, it will probably have something to do with the fact that all norms are equivalent, but I couldn't find a proof.
The answer to both is "no".
For the second question, consider the maximum norm on $\mathbb{R}^2$, $\lVert (x,y)\rVert_{\infty} = \max\: \{ \lvert x\rvert, \lvert y\rvert\}$, the subspace $U = \{ (0,y) : y \in \mathbb{R}\}$ and $a = (1,0)$. Then $\lVert (0,y) - a\rVert_{\infty} = 1$ for all $y \in [-1,1]$, and $\lVert u - a\rVert_{\infty} \geqslant 1$ for all $u \in U$.
For the first question, also consider $\mathbb{R}^2$, and the two norms $\lVert (x,y)\rVert_v = \max\: \{ 10 \lvert x\rvert, \lvert y\rvert\}$ and $\lVert (x,y)\rVert_h = \max \{ \lvert x\rvert, 10\lvert y\rvert\}$. Let $U = \{(x,x) : x \in \mathbb{R}\}$ and $a = (1,0)$. Then
$$\lVert(x,x) - (1,0)\rVert_v = \max \: \{ 10 \lvert x-1\rvert, \lvert x\rvert\}$$
is minimised for $x = \frac{10}{11}$, and
$$\lVert (x,x) - (1,0)\rVert_h = \max \: \{ \lvert x-1\rvert, 10 \lvert x\rvert\}$$
is minimised for $x = \frac{1}{11}$.