Projection onto Basis Has Lower Error

Question

Let $\{x_i\}_{i=1}^n$ be a linearly independent subset of a separable but infinite-dimensional Banach space $X$ with norm $\|\cdot\|_X$, and let $y \in X-\{x_i\}_{i=1}^n$. Then if each $x_i$ is non-zero is it true that $$ \min_{k_1,\dots,k_n \in \mathbb{R}}\|y-\sum_{i=1}^n k_ix_i\|_X<\|y -x_1\|_X? $$

Answer 1

$\leq $ is obvious but strict inequaity may not be true. Consider the case when $%X$ is Hilbert space, $n=1$ and $y=x_1+z$ with $z$ orthogonal to $x_1$. Then the square of LHS becomes the infimum of $(1-k_1)^{2}\|x_1\|^{2}+\|z\|^{2}$. Clearly this infimum is $\|z\|^{2}$ which is same as the square of RHS.