Give X is a Banach space and the dual space of X is $X^* = L(X, \mathbb{R})$. Suppose that $x, y \in X$ such that $T(x)=T(y)$ for all $T \in X^*$. This is a question: $x=y$ or $x\ne y$?
I have tried calling $x_n \to x, y_n\to y$, so we have $T(x_n) \to T(x)$ and $T(y_n)\to T(y)$. But I still can't solve it. And now, I don't know what I should start from where?
Please help me, I would like to receive some feedback! Thanks so much!
If $x$ and $y$ are linearly independent we can define continuous linear functional $f$ in $M=span \{x,y\}$ by $f(ax+by)=a$. By Hahn Banach Theore we can extend $f$ to an element $T$ of $X^{*}$ and we have $T(x)=f(x)=1$ and $T(y)=f(y)=0$, so $Tx\neq Ty$.
Conclusion. $x$ and $y$ have to be linearly dependent. Suppose $y=ax$. Define $f$ on $span \{x\}$ by $f(cx)=c$ and apply Hahn Banach Theorem again. Conclude that we must have $a=1$ so $x=y$.