This is from an example of a textbook that I didn't totally understand.
Let's suppose $V$ is the vector space of all polynomials functions from $\mathbb R$ into $\mathbb R$ which have degree less than or equal to 2.
Let $t_1$, $t_2$ and $t_3$ three distinct real numbers and let $$L_i(p) = p(t_i)$$
Then $L_1$, $L_2$ and $L_3$ are linear functionals on $V$ and they are linear independent.
Now, for proving that statement I put $$ 0 = c_1L_1+c_2L_2+c_3L_3 $$ for all $c_1$, $c_2$ and $c_3$ and for each $p$ in $V$, and this should give us $c_1 = c_2 = c_3 = 0$
How can I prove that for all $p$ in $V$? I was thinking on considering a basis for the polynomials but then I'll introduce new scalars different from $c_1$, $c_2$ and $c_3$ and I got things messed up. Thanks in advance.