The classical definition of a function \( f : X \rightarrow Y \) that built our \( L^p \)-spaces had a caveat. An almost everywhere equivalence class of functions was needed for compatibility with the integral norms of our Banach space and its dual. However, this framework has limitations directly describing derivatives of æ differentiable classical functions. The objects arising from differentiation, like the Dirac delta, can instead be described by a theory of distributions.

Rather than assigning values to almost all points in an open set \(\Omega \subset \mathbf{R}^d\), a distribution \(F\) is characterized by its action on smooth functions \(\varphi\). When \(f \) is a locally integrable classical function on \(\Omega\), its associated distribution \( F_f \) is the map defined by \[ F (\varphi ) = \int_{\mathbf{R}^d} f(x)\varphi(x)dx, \] over all appropriate test functions \(\varphi \). Observe that if \( f \) is sufficiently differentiable, integration by part yields \[\int_{\mathbf{R}^d}\left(\partial_{x}^{\alpha} f\right) \varphi d x=(-1)^{|\alpha|} \int_{\mathbf{R}^d} f\left(\partial_{x}^{\alpha} \varphi\right) d x + \int_{\partial\mathbf{R}^d} f \varphi d x. \] Our goal is to define distributional derivatives accordingly, so we must choose the properties of our test functions to suit this end.

Test functions

We want our space of distributions to capture differentation for all locally integrable functions. Distributions themselves should be infinitely differentiable in the sense of distributions, so we require our space of test functions \(\mathcal{D} \) to constitute only smooth functions. Further, the boundary terms from integration by parts should vanish so we require each test function \(\varphi \) to vanish at infinity. For the larger class of distributions, this is achieved by requiring each \(\varphi\) to have compact support.

For a distribution is \( C_0^\infty(\Omega)\), the space of smooth functions compactly supported on \(\Omega\) We want our The space of test functions denoted \( \mathcal{D}(\Omega) \)