Main idea: usual functions can be reinterpreted like acting over an special set of functions: the test functions (functions with compact support). So we can think of functions like something similar to covectors. But there are more operators than the usual functions: they are the distributions.
Let us define $D(\mathbb{R}^n)$ as the set of $\mathbb{C}^{\infty}$-functions with compact support. We call a distribution to every continuous linear transformation:
$$ T:D(\mathbb{R}^n)\mapsto \mathbb{R} $$Usually, the action will be denoted, for $\varphi \in D(\mathbb{R}^n)$:
$$Any locally integrable function $f:\mathbb{R}\mapsto \mathbb{R}$ define a distribution $T_f$ in the following way
$$An important example of distribution, not really coming from a function, is the Dirac delta function $\delta$ (not really a function!)
About the nature of distributions as sheafs: see this.
We also can define the derivative of a distribution. It would be convenient that for usual functions $(T_f)'=T_{f'}$, that is
$$ <(T_f)',\varphi>=\int _ { \mathbf { R }^n } f' ( x ) \varphi ( x ) d x= [ f ( x ) \varphi ( x ) ] _ { - \infty } ^ { \infty } - \int _ { \mathbf { R }^n } f \varphi ^ { \prime } d x = - \left\langle f , \varphi ^ { \prime } \right\rangle $$where we have integrated by parts.
So in general we define $T'$ as
$$ \langle T',\varphi \rangle=-\langle T,\varphi' \rangle $$From this point of view it can be check that the Dirac delta is the derivative of the Heaviside function
$$ H (x) = \left\{ \begin{array} { l l } { 0 , } & { x < 0 } \\ { \frac { 1 } { 2 } , } & { x = 0 } \\ { 1 , } & { x > 0 } \end{array} \right. $$Attention: value at 0 could be controversial.
At the same way that Dirac delta means evaluation at 0 of a test function, we can interpret that general distributions are "generalized points", or "matter distribution", or "bodies"; and the action over a function is "evaluate the function over that body". If the distribution have compact support I think we can choose any function, not only test functions.
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Author of the notes: Antonio J. Pan-Collantes
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