Structure/Order of explaining distribution-theory

Question

please tell me if this is too off-topic, then I will delete the post. In my master thesis I want to use (and explain) distributions in order to solve some PDE, but since covid still is present it is hard for me to get academic feedback, so I thought maybe you guys can have a quick look and tell me whether the following structure is fine for introducing distributions. My global structure will be something like this:

• Introduction: PDEs are everywhere and solving them is hard
• Main part
  • PDEs
  • Distributions
  • Fourier Transforms
  • The Laplace Equation
  • The Wave Equation
  • A hypoelliptic operator of fourth order
• Conclusion: PDEs are still hard, but with distributions and fourier transforms specific problems can often be solved and surprising facts be derived

Now for Distributions I wanted to introduce it with "everything can be differentiated" and "Then for example the divergence theorem reduces to calculating the derivative of a characteristic function." The structure should be like:

• Derivative of the Heaviside function
  • A first example.
  • There is no weak derivative for $\mathcal{H}$, but we want to have something like $\delta$.
• Test functions
  • There are functions in $\mathcal{C}_c^\infty$, for example the standard mollifier.
  • Also $lim_{\epsilon \rightarrow 0} \eta_\epsilon = \delta$
  • $\phi \mapsto \phi^{(k)}(0)$ should be a distribution, so there needs to be a topology on $\mathcal{C}_c^\infty$ that respects the derivatives.
  • $\mathcal{D}$ has the topology induced by $\sup_{x \in K} sup_{\alpha \in \mathbb{N}^n} \partial^\alpha \phi$.
  • This is not metrizable.
• Definition of Distributions
  • $\mathcal{D}'$ is the dual space of $\mathcal{D}$.
  • One can easier work with the usual definition of a linear map T with $|T(\phi) \leq C \sum_{|\alpha| \leq k} sup_{x \in K} |\partial^\alpha \phi(x)|$.
  • Operators can be defined by its behaviour on test function, for example: $<\partial^\alpha T ,\phi> = <T , (-1)^\alpha \partial^\alpha \phi >$
  • Every distribution is the limit of a series of Testfunctions
• Support of a Distribution
  • One can define the support of a distribution
  • A distribution on $ X \subset \mathbb{R}^n$ can be extended to $\mathbb{R}^n$
  • We can restrict Distributions to a subset
  • Multiplication of Distributions
  • Definition of singsupp
• Operators on Distributions
  • The Tensor product
  • Convolution with a function
  • Convolution in general
• Working with Distributions
  • if $T' = 0$ then T is constant
  • calculating some derivatives
  • Complex Distributions
  • Calculating some limits
• Fundamental solutions
  • Definition
  • Application of $P (E_P * f) = f$ and $ u = E_P * (P u)$

I'm aware that this is somewhat unstructered. But this just reflects that I can't differ between technicalities and important theorems, apart from that fundemantal solutions are important for solving the inhomogeneous equation.

Thanks already and as I said, if it is too off-topic I will delete the question.