Bounded linear functionals in solving PDE Many theorems in functional analysis, Rietz, Hahn-Banach, for example, are used to find linear functionals in certain spaces.
But why are bounded linear functionals useful in solving PDE?
This question may sound too general and philosophical. But any explanation is welcomed!
 A: You can think of a linear functional on a vector space as giving you a coordinate in that space. A continuous linear functional is a coordinate that varies continuously as the vector varies continuously; one is not so much interested in the coordinates that are disconnected from the topology of the space.
Representation of a linear functional gives the existence of something such as a function or measure on a space, and that object can be used to solve equations, or to represent another object. For example, if $A$ is a continuous selfadjoint operator on a Hilbert Space, and $x$ and $y$ are fixed vectors, then
$$
                  p \mapsto (p(A)x,y)
$$
is a continuous linear functional from polynomials $p\in C[a,b]$ into the scalar field. So this function extends continuously to all of $C[a,b]$. And we know that such linear functionals are represented by integration with respect to a complex measure $\mu_{x,y}$:
$$
                       (p(A)x,y) = \int_{a}^{b}p(t)d\rho_{x,y}(t).
$$
Out of this you can get the spectral theorem for selfadjoint operators on a Hilbert space, which is a useful way to represent a general selfadjoint linear operator.
If you're working in a space such as $L^{2}[a,b]$, every continuous linear functional $\Phi$ has a representation
$$
                   \Phi(f) = \int_{a}^{b}f(t)g(t)dt
$$
for some $g \in L^{2}[a,b]$. You can use this representation to show that various types of differential equations have weak solutions, which is even more useful when applied to a region of $\mathbb{R}^{n}$. The existence of a particular type of representation becomes a way of solving equations by establishing the existence of a function.
This is all part of Functional Analysis.
