Existence of a function so an integral becomes null. Let E be an interval of $\mathbb{R}$ for example $E=[a,b]$. Let $A(s) \in \mathbb{R}^n$ a vector which components are real functions defined in E. How can I proof that there is a non-null real function $\beta (s)$ defined in E such that the integral:
$$
\int_E A(s) \beta (s) = 0
$$
is null? I think this is true in general, but I am not sure. If it isn't, what conditions are required for $A(s)$ for this to be true?
 A: I assume that you want the coordinate functions of $A=(A_1, \dots, A_n)$ to be in $L^2(E)$. Then what you want is that
$$ \langle A_j, \beta \rangle =0 \qquad \forall j\in \{ 1, \dots, n\}. $$
So you are asking whether the orthogonal complement of $span\{A_1, \dots, A_n\}$ is nontrivial. This is indeed the case. You can see this as
$$ L^2(E)= span\{A_1, \dots, A_n\} \oplus span\{A_1, \dots, A_n\}^{\perp}.$$
However, $L^2(E)$ is infinite dimensional and $span\{A_1, \dots, A_n\}$ has dimension $n$, thus $span\{A_1, \dots, A_n\}^{\perp}$ is infinite dimensional (and therefore nontrivial).
I would guess that one can find weaker assumptions than $L^2(E)$, but I presumably then the proof gets more complicated.
A: If you consider the component-functions of A, which I will call $a_1, \dots, a_n$, you are searching for a function $\beta$, such that $\int_E a_i(s) \beta(s) \mathrm d s=0$ for all $i \in \mathbb N$. But the integral $\int_E a_i(s) \beta(s) \mathrm d s=0$ is just the scalar-product of the Hilbert space $L^2(E)$, so you are searching for a function, which is orthogonal with regard to this scalar-product to all $a_i$, which clearly exists, since the subspace spanned by the $a_i$ is finite.
If you know what a Hilbert space is, this should be a good enough answer, else I will try to elaborate more one this.
