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Suppose I have a class of functions $\mathcal{F}$ with the property that

$\int f(x) g(x) = \int f(x) h(x)$ for all $f \in \mathcal{F}$ implies $g = h$.

What's the correct name for this property? If $g$ and $h$ are in $L^p$, do I say that $\mathcal{F}$ is a seperating set, or that it seperates points in $L^p$, or something else?

If $\mathcal{F}$ is the class of smooth functions with compact support and $g$ and $h$ live in $L^p$, is the implication correct? If so, what's that result called?

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  • $\begingroup$ If we talk about $L^p(\mathbb R^n)$ with $1\leq p<\infty$, then the result is true because the closure for the norm $L^p$ of the class of smooth functions with compact support is $L^p(\mathbb R^n)$. It's also true for $L^p(\mathcal O)$, where $\mathcal O$ is an open subset of $\mathbb R^n$. $\endgroup$ Jul 15, 2011 at 12:28
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    $\begingroup$ Yes, you could say "$\mathcal F$ separates $L^p$". This is true in particular if the span of $\mathcal F$ is dense in the dual $L^q$. $\endgroup$
    – GEdgar
    Jul 15, 2011 at 13:10
  • $\begingroup$ I think statement of this form often go under the name "fundamental lemma of the calculus of variations". $\endgroup$
    – Dirk
    Jul 15, 2011 at 13:45

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The function $g\mapsto \int fg$ is a linear function. Depending on the topology you impose, it is a continuous linear function. What you are seeing is that $\mathcal F$ has as its closed linear span the entire space (in your case $L^p$).

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