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I'm teaching Distribution theory and many of my students still believes that there is only one kind of distribution : the distribution that can be identified to a $L^1_{\text{loc}}$ function. And I know that there is plenty of examples of distributions that can not be represented with a $L^1_{\text{loc}}$ function. But when I say plenty, can we say how big the subspace of distributions that comes from a $L^1_{\text{loc}}$ function is in the space of all distributions ? I don't know if there is a way to mathematically measure this, if not, just tell me that there is no easy way to compare this two spaces and why.

user37238
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    I think it's easy to prove that $\dim(D'(\Bbb R)/L^1_{loc}(\Bbb R))=\infty$. And, after all, they must know about $\delta_0$ - a distribution which can not be represented by a $L^1_{loc}$ function. – TZakrevskiy Dec 19 '13 at 18:08
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    On the other hand, the "very small" subspace $C_c^\infty$ taken as distributions, is weakly or even strongly dense in $\mathcal{D}'$. – Vobo Dec 19 '13 at 21:25
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    @Vobo Dense in which sense? – user37238 Dec 20 '13 at 11:16
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    @user37238 Dense in the topologies $\beta(\mathcal{D}',\mathcal{D})$ or $\sigma(\mathcal{D}',\mathcal{D})$, the strong or weak(*)-topology on $\mathcal{D}'$, the topology of uniform convergence on the bounded subsets of $\mathcal{D}$ or the topology of pointwise convergence. – Vobo Dec 20 '13 at 14:20

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