$f_1,...,f_n$ be linear functionals on a real vector space $V$, then is there a norm on $V$ which makes every $f_i$ continuous?

Question

Let $V$ be a real vector space, $f_1,...,f_n$ be linear functionals on $V$; then does there exist a norm on $V$ with respect to which each of $f_i$ is continuous? And what if we have infinitely many, linearly independent, such functionals?

Just in general, don't add spaces between the end of a word and punctuation. For example, it should be "...such functionals?" and not "...such functionals ?". — , Apr 29 '16 at 21:20

score 3 · Accepted Answer · edited Apr 13 '17 at 12:21

The answer for infinitely many functionals is no; there's a counterexample here.

For finitely many functionals it must be yes...

Right. First, there is a norm on any real vector space $X$, for example if $B$ is a (Hamel) basis define $$\left\vert\left\vert\sum_{b\in B}c_b b\right\vert\right\vert=\sum_{b\in B}|c_b|.$$Now if $||\cdot||$ is a norm on $X$ define a new norm by $$|||x|||=||x||+\sum_{j=1}^n|f_j(x)|.$$

$f_1,...,f_n$ be linear functionals on a real vector space $V$, then is there a norm on $V$ which makes every $f_i$ continuous?

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