Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)>\alpha$. That is, the distance from the unit sphere of $\langle x \rangle$ (where $\langle A\rangle$ denotes the linear span of $A$) to $Y$ is at least $\alpha$.

I'm looking for a proof or counterexample for the following variation of the Riesz Lemma:

Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, there is $x\in X$ such that the distance from $\langle x\rangle$ to the unit sphere of $Y$ is at least $\alpha$.

I am mostly interested in positive answers, so partial positive answers would be nice if the result is false. Particularly, I'm only concerned with Banach spaces, so this can be assumed without problem.

EDIT: Pietro presented a clever construction in finite dimension and with $\mathrm{codim}(Y)=1$. Is there also a counterexample when $\dim(X)=\infty$ and $\mathrm{codim}(Y)=\infty$?

`\|x\|`

, looks different from $||x||,$ and $\|x\|\|y\|,$ coded as`\|x\|\|y\|`

, looks very different from $||x|| ||y||,$ coded as`||x|| ||y||`

. The former notation is standard usage. $$ \begin{align} & ||x|| ||y|| \\ {} \\ & \|x\|\|y\| \end{align} $$ $\endgroup$`\lVert\rVert`

can work even better than`\|\|`

. Compare, for example,`\|-x\|`

$\|-x\|$ to`\lVert-x\rVert`

$\lVert-x\rVert$. Although a bit awkward, this can be swept up in a macro like`\newcommand\norm[1]{\lVert#1\rVert}`

, and even repeated such macro definition can be semi-automated using`\DeclarePairedDelimiter`

from mathtools. $\endgroup$