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I am self-studying P. Lax's functional analysis book. Here is an exercise in p123. it is supposed to be very easy, but I really couldn't see it.

Could you anyone help me out? Thanks.

Let $\{ l_\alpha\}$ be a collection of linear functions in a linear space $X$ over $\mathbb{R}$ that separates points; that is, for any two distinct points $x$ and $y$ of $X$ there is an $l_\alpha$ such that $l_\alpha(x) \neq l_\alpha(y)$.

Assertion: A linear functional $l$ is continuous in the topology generated by $\{ l_\alpha\}$ iff it is a finite combination of the $l_\alpha$.

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    @t.b. : What does $|l_i(x)|\le \epsilon$ mean ?is it not true that $l(x)$ is a discrete point ? Can you help me to get rid of confusion . Thank you .2012-12-01

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