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If I understand correctly, distributions $F_n \in C^\infty_c(\mathbb{R})^*$ are defined based on how they act on test functions $\phi \in C^\infty_c(\mathbb{R})$.

What does it mean then to say $F_n \rightarrow F$ in $L^p(\mathbb{R})$?

(I cannot take it to mean $\langle F_n, \phi \rangle \rightarrow \langle F, \phi \rangle$ since this is just the convergence of numbers and has nothing to do with $L^p$ convergence.)

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up vote 1 down vote accepted

If $F$ is a function is $L^p$, it's in particular locally integrable, so it defined a distribution by the formula $$\langle F,\varphi\rangle:=\int_\Bbb R F(x)\varphi(x)dx,\varphi\in\cal C_c^\infty(\Bbb R).$$ So $F_n\to F$ in $L^p$ means that the representing functions converge in $L^p$. It's weel defined because if $F\in L^p$ is such that $\int_\Bbb R F(x)\varphi(x)dx=0$ for all $\varphi\in\cal C_c^\infty(\Bbb R)$ then $F$ is the equivalence class of the null functions for equality almost everywhere.

Note that not every distribution can be represented by a $L^p$ function (it's even not true replacing $L^p$ by locally integrable, as $\delta_0$ shows). Furthermore, convergence in $L^p$ implies convergence in $(\cal C_c^\infty(\Bbb R))^*$.

All of the previous work can be done in $\Bbb R^d$, with $d$ integer $\geqslant 1$.

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