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We work with real Banach spaces. Let $X$ be an infinite-dimensional Banach space and let $(f_n)$ be a sequence of norm-one functionals on $X$ that are linearly independent. Can we find an element $x\in X$ such that $f_n(x)>0$ for all $n$?

I guess not. What if $X$ is reflexive?

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  • Let $F_{n}={ x : f_{n}(x) \le 0 }$ for $n=1,2,3,\cdots$. Then $F_{n}$ is closed. You want to find $x \in X\setminus\bigcup_{n}F_{n}$ because that means $x \notin F_{n}$ for all $n$ and, hence, $f_{n}(x) > 0$ for all $n$. Have you thought about what happens if there is no such $x$? – Disintegrating By Parts Feb 27 '15 at 20:00
  • http://math.stackexchange.com/questions/313425/a-certain-element-which-makes-functionals-positive – Tomasz Kania Mar 01 '15 at 15:06

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