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1. Test Function

The space of test function is $D (R^{n}) = C_{c}^{\infty} (R^{n})$ equipped with the notion of convergence.
- $C_{c}^{\infty} (U)$ with $U \subset R^{n}$ , is a dense subset of $C^{\infty} (U)$ , such that for all $f \in C_{c}^{\infty} (U)$ have compact .

$φ_{k} \overset{D}{\to} φ$
$\exists bounded set M : \forall k, x \notin M ⟹ φ_{k} (x) = 0.$
$φ_{k} \to φ unifromly$ and $\forall α, D^{α} φ_{k} \to D^{α} φ uniformly$ where $α$ is the multi-index.

A distribution $T$ on $U$ is a linear functional on $C_{c}^{\infty} (U)$ , that is continuous when $C_{c}^{\infty} (U)$ is given a topology called the canonical LF topology.
$T : D (R^{n}) \to R$ with
- Linearity
- Continuity
  - Sequentially continuous with respect to the convergence of the test functions. $φ_{k} \overset{D}{\to} φ ⟹ T (φ_{k}) \to T (φ) .$

Product of two distributions is also a distribution: $(f \cdot S) (φ) = (T_{f} \cdot S) (φ) := S (f \cdot φ)$ where $S \in D^{'} (R^{n}), f \in C^{\infty} (R^{n})$ .
Equivalently: $⟨ f \cdot S, φ ⟩ := ⟨ S, f \cdot φ ⟩ .$
Motivated by $T_{f} \cdot T_{g} = T_{f \cdot g}$ .

For a invertible linear map $A : R^{n} \to R^{n}$ : $⟨ T \circ A, φ ⟩ := ⟨ T, | det A |^{- 1} φ \circ A^{- 1} ⟩ .$
With different notation: $⟨ T (A x), φ (x) ⟩ := ⟨ T (x), | det A |^{- 1} φ (A^{- 1} x) ⟩ .$

Extension of .
$⟨ ψ * T, φ ⟩ := ⟨ T, \overset{ˇ}{ψ} * φ ⟩$ where the check operator simply reverses the argument: $\overset{ˇ}{ψ} (x) = ψ (- x)$ .
The convolution $* : D (R^{n}) \times D^{'} (R^{n}) \to D^{'} (R^{n})$ is a bilinear map, which can be thought of as a multiplication.
Notably, $ψ * δ = ψ .$

$T (x)$ , although not correct, may be used to denote the variable that the distribution is evaluated by.

For every compact subset $K \subseteq U$ , there exists constants $C > 0$ and $N \in N$ such that for all $f \in C_{c}^{\infty} (U)$ with support contained in $K$ : $| T (f) | \leq C sup {| \partial^{α} f (x) | : x \in U, | α | \leq N} .$
For every compact subset $K \subseteq U$ an every sequence ${f_{i}}_{i = 0}^{\infty}$ in $C_{c}^{\infty} (U)$ whose supports are contained in $K$ : If ${\partial^{α} f_{i}}_{i = 1}^{\infty}$ converges uniformly to zero on $U$ for every multi-index $α$ , then $T (f_{i}) \to 0$ .

Distribution can be added and scalar multiplied.
Distributions forms a equipped with the bilinear map: $⟨ \cdot, \cdot ⟩ : D^{'} (R^{n}) \times D (R^{n}) \to R (or C) .$

A distribution $T \in D^{'} (R^{n})$ is regular, if $\exists f \in L_{1, l o c} (R^{n}) : T (φ) = \int_{R^{n}} f (x) φ (x) d x .$
- $L_{1, l o c} (R^{n})$ is the .
- Dirac Delta Distribution is not regular.

For a distribution $T \in D^{'} (R^{n})$ , the distributional partial derivative $D^{α} T \in D^{'} (R^{n})$ of $T$ is: $⟨ D^{α} T, φ ⟩ = (- 1)^{| α |} ⟨ T, D^{α} φ ⟩$ where $α$ is the multi-index.
Motivated by $⟨ T_{f^{'}}, φ ⟩ = - ⟨ T_{f}, φ^{'} ⟩ .$
The differential operator $D^{α} : D^{'} (R^{n}) \to D^{'} (R^{n})$ is linear and continuous.
- $D^{α} (\sum_{k = 1}^{\infty} T_{k}) = \sum_{k = 1}^{\infty} D^{α} T_{k} .$
- The linearity is enough for finite sum, but continuity is required for the infinite sum.
The of a differential equation can be obtained as a distribution.
For all multi-indices $α$ : $D^{α} (ψ * T) = (D^{α} ψ) * T = ψ * (D^{α} T) .$
Given a fundamental solution $P (D) E = δ$ , the partial differential equation $P (D) u = f$ is solved by $f * E$ , as long as $f$ is one of the test function.
- See