Distributions

Distributions are continuous linear applications on different test functions.
Usually test functions are in the space $ \mathcal{C}_{0}^{\infty}(R^n) $ with topology $ \mathcal{T} $ defined by convergent sequence, as follows
A sequence $ (\phi_n)_{n\in N},\phi\in\mathcal{C}_{0}^{\infty}(R^n) $ is convergent to $ \phi $ if and only if there is a compact $ K\subset R^n $ such as for all $ n\in N,supp \phi_n\subset K $ and for all $ \alpha=(\alpha_1,\alpha_2,...\alpha_n) $,

$$\partial^{\alpha}\phi_n=\frac{\partial^{|\alpha|}\phi_n}{\partial x_1^\alpha_1\partial x_2^\alpha_2...\partial x_n^\alpha_n} \:is \:uniform \:convergent \:to \:\:\partial^{\alpha}\phi$$

We write $ \mathcal{D}(R^n) $ for the space $ (\mathcal{C}_{0}^{\infty}(R^n),\mathcal{T}) $
There is a useful equivalent condition for a linear map on $ \mathcal{C}_{0}^{\infty}(R^n) $ to be a distribution:
Theorem. A linear map u on $ \mathcal{C}_{0}^{\infty}(R^n) $ is a distribution if and only if
$ \forall K\subset R^n $, there are constants $ M>0 \:and \: k\in N  $such as $ \forall \phi\in\mathcal{D}_K(R^n) $ we have

$$|<u,\phi>|\leq M \sum_{|\alpha|\leq k}sup_{x\in K}|\partial^{\alpha}\phi(x)|$$

The minimal integer k in this definition is called the order of distribution on K.
An important example of distribution is Dirac $ \delta_{a} $ defined by

$$<\delta_{a},\phi>=\phi(a)$$

Dirac $ \delta_{a} $ is a distribution of order zero.

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