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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.

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    $\begingroup$ What if $X=\mathbb{R}^2$, $Y=(y=0)=\{(x,0)\}$, $y=(1,0)$? Any extension $x^*$ looks like $(1,t)$ for some $t$, and then $\{x \in X \colon x^*(x)=1\} = \{(a,b) \colon a+bt=1\}$, not contained in $Y$. $\endgroup$
    – Zach Teitler
    Commented Jun 10, 2019 at 19:46

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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 $t\ne 0$.

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