Let $V$ be the $\mathbb{R}$-Vector space of polynomials with degree $\leq$ 2. Let $\psi\colon V \times V \rightarrow \mathbb{R}$ be the bilinear form $$(p,q) \mapsto \int_{0}^{1}p(x)q(x)dx$$
Let $\phi$ be a linear form $p \mapsto p(0)$. Find a $q \in V$, such that $\phi(p)=\psi(p,q)$ for all $p\in V$
I still have trouble understanding, what this $q$ that I'm looking for actually is and especially how I would manage to find it.
Best Answer
You are looking for a polynomial $q(x) \in V$ such that $$ \int_0^1 p(x) q(x) = p(0) $$ for all $p(x) \in V$.
The element $q \in V$ is determined by its coefficients: $q(x) = b_0 + b_1 x + b_2 x^2$, for some $b_0, b_1, b_2 \in \mathbb{R}$. Using some clever choices of $p(x)$ you should be able to determine equations satisfied by those coefficients.