Let $V=\mathbb{P}_1[x]$ Let $\alpha_{1}(p)=\int_0^1{p(t)}dt$ and $\alpha_{2}(p)=\int_0^2{p(t)}dt$ for each $p \in V$
- Show $\{\alpha_{1},\alpha_{2}\}$ forms a basis for the dual space
- Find basis $\{p_1(t)p_2(t)\}$ s.t. $\{\alpha_{1},\alpha_{2}\}$ is the dual basis of $\{p_1(t)p_2(t)\}$
- Assume $V$ has the inner product defined by $\langle p,q\rangle=\int_0^1{p(t)q(t)}dt$ Find polynomial s.t. $\alpha_2(p)=\langle p,q\rangle$
For 1, since the basis for $V$ is $\{1,x\}$, then evaluating for each function gives non-zero scalars, for the sum of which can only be $0$ iff the functional values have scalar coefficients of zero.
For 2, is this merely a computational question?
For 3, not sure.
EDIT: I think I figured out that the polynomial for $3$ is $t + \frac{1}{2}$