I have a homework, I have tried so much on this question, but I couldn't solve it. How can I prove it? Can you help me? Thank you.
$E$ is a normed space and $E'$ is the topological dual of $E$. If $\phi_1,...,\phi_n\in E'$ and $\epsilon>0$, then
$U(0;\phi_1,...,\phi_n;\epsilon)$=$\{x\in E:|\phi_j(x)|<\epsilon,1\leq j\leq n\}$ which is neighborhood of $0$ on $(E',\sigma(E',E))$ weak topology, includes a following neighborhood:
$U(0;\psi_1,...,\psi_n;\delta)$=$\{x\in E:|\psi_j(x)|<\delta, \psi_1,...,\psi_n\in E' ~~~\textrm{linear independent}~~~\delta>0,1\leq j\leq n\}$