Suppose that $(X, \|\cdot\|\_X)$, $(Y, \|\cdot\|\_Y)$ are two Banach spaces such that $X\subset Y$ and $\|x\|\_Y\leq \|x\|\_X$ for all $x\in X$ and $X$ is dense in $(Y, \|\cdot\|\_Y)$. 


Every functional $y^\*\in Y^\*$ also yields a functional $y^\*|\_X\in X^\*$ with $\|y^\*|\_X\|\_{X^\*}\leq \|y^\*\|\_{Y^\*}$. Suppose that $(y^\*\_n)\_{n=1}^\infty$ is a sequence in $Y^\*$ such that $\lim\_n \|y^\*\_n\|\_{Y^\*}=\infty$ and $\|y^\*\_n|\_X\|\_{X^\*}=1$ for all $n$. Can we find a sequence $(x\_n)\_{n=1}^\infty\subset B\_X$ such that $\inf\_n |y^\*\_n(x\_n)|>0$ and $\lim\_n \|x\_n\|\_Y=0$? 


If the answer is no in general, is it true if $A\subset B$ are subsets of $c\_{00}$ (the finitely non-zero scalars sequences), $$\|x\|\_X=\sup\{|\langle f,x\rangle|: f\in B\}$$ and $$\|y\|\_Y=\sup \{|\langle f,y\rangle|: f\in A\},$$ and $X,Y$ are the completions of $c\_{00}$ with respect to these norms? 



Norming functionals for vectors in intersections
Let $X=\ell^1$, $Y=\ell^2$. Take $y\_n^\*=(1,1/n,1/n,\dots,1/n,0,0,\dots)$ with $\frac 1n$ repeated $n^3$ times. Looks like we are fried, or am I missing something in the setup?


