# Tagged Questions

Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

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### About test functions for supersolutions

Let $B_{1}$ the unit open ball in $\mathbb{R}^{n}$ and $u \in H^{1}(B_{1})$. For $k,m >0$, let $\bar{u} = u^{+} + k$ and $\bar{u}_{m} = \bar{u}$ if $u < m$ and $\bar{u}_{m} = k+m$ if $u \geq m$. ...
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### Do tempered distributions form a topological subspace of the space of distributions?

I'm learning about distributions and tempered distributions. From what I understand, by "enlarging" the space of test functions $\mathcal{D}$ to the Schwarz space $\mathcal{S}$ and correspondingly "...
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### $\Lambda\psi$=0 if $(D^{\alpha}\psi)(x)$ for every $x\in$ supp $\Lambda$ and every multi index $\alpha$

I can show that $\Lambda\psi$=0 if $(D^{\alpha}\psi)(x)$ for every $x\in$ supp $\Lambda$ and every multi index $\alpha$ given the support of $\Lambda$ is compact. But how one extend this argument for ...
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### Continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is a distribution.

Prove that every continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is of the form $\Lambda\mapsto\Lambda f$ for some distribution $\Lambda$ with compact support. I am stuck at this ...
### Positive distribution $\Lambda$ as positive Radon measure
Exercise 4 of Chapter 6 in Rudin's Functional Analysis states that every "positive" distribution $\Lambda\in D^{'}(\Omega)$, i.e, $\Lambda\psi\geq 0$ whenever $\psi\in D(\Omega)$, is a positive ...