Suppose $Y$ is a subspace of a normed linear space $X$ and let $y\in S\_Y, y^\*\in S\_{Y^\*}$ such that $y^\*(y)=1$, where $S\_Y$ denotes the closed unit sphere in $Y$. My question is the following:


Is it possible to find an extension $x^\*$ of $y^\*$ to $X$ such that $$\{x\in X:x^\*(x)=1\}\subset Y.$$ Any suggestion will be thankfully appreciated.



Can a hyperplane be contained in a subspace?
The answer is "No, unless $X=Y$". 


In fact, if there is $u\in X\backslash Y$, then the vector $v=u-x^\*(u)y$ is not in $Y$ and satisfies $x^\*(v)=0$. Therefore all vectors $y+tv, t\ne 0$ satisfy $x^\*(y+tv)=1$, but $y+tv$ are not in $Y$ for $t\ne 0$.


