# Convergence in generalized functions(Distributions)

I know that convergence in $D'(\Omega)$ (The space of distributions) is just the convergence in $\mathbb{R}$ for every test function $\phi$. But I am not sure about something like " Norm convergence implies convergence is distributions" , what about $L^{p}$ convergence implies convergence in distributions and $L_{loc}^{2}$ etc.A sequence $T_{n}$ of distributions convergence to another distribution $T$ if we have $\lim_{n \rightarrow \infty} \langle T_{n},\phi \rangle = \langle T, \phi \rangle$ for ever test function $\phi$. I appologize if it seems something obvious. Any discussion will be highly appreciated.

• Use Hölder's inequality and $\phi\in L_q$ ($1/p+1/q=1$)
– Dap
Sep 29, 2017 at 5:27
• Not clear to me . Can you give a precise idea if what do you mean ? Sep 29, 2017 at 7:03

To every function $u\in L^p(\Omega)$ you can associate a distribution $T_u$ defined by $$T_u(\phi)=\int_\Omega u\phi\, dx.$$ If you now have a sequence of functions $u_n$ in $L^p(\Omega)$ converging to some function $u$ in $L^p(\Omega)$, and you consider the corresponding distributions $T_{u_n}$, then as Dap said, using Hölder's inequality you get $$|T_{u_n}(\phi)-T_{u}(\phi)|=\left|\int_\Omega (u_n-u)\phi\, dx\right|\le \Vert u_n-u\Vert_{L^p}\Vert \phi\Vert_{L^{q}}\to 0$$ as $n\to\infty$. Thus $T_{u_n}$ converges to $T_{u}$ in the sense of ditributions. Since each $\phi$ has compact support, it would be enough to assume that $u_n$ converges to $u$ in $L^p(K)$ for every compact set $K\subset\Omega$.
• Or integrate $u_n \to u\in L^1_{loc}$ to obtain a locally uniformly convergent sequence of continuous functions, and say that convergence in $C^0$ implies convergence of all the distributional derivatives. Oct 1, 2017 at 2:26
• @Mathslovershah If $u_n \in L^p, p \ge 1$ then $v_n(x) = \int_0^x u_n(y)dy$ is continuous and $v_n \to v$ locally uniformly, which implies it converges in the sense of distributions and the same for all the distributional derivatives $v_n' = u_n \to u = v'$ Oct 2, 2017 at 15:02