# Functional analysis

## Differential equations and Fourier transformation

Differential equations describe the change of state. The change relates to the state. The solutions of the differential equations are the status equations. The initial conditions set the time $$t$$ and status $$y$$. The boundary conditions are the value of boundary $$y_0$$ and $$y_1$$. $$dy \over dt$$ $$= ay + q(t)$$ starting from $$y(0)$$ at \$t=0. inital conditions $$t = 0$$ and $$y=1$$ $$q(t)$$ is a input and $$y(t)$$ is a response.

## Positive Borel measures

This is a note of real and complex analysis chapter 2. Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear functionals, not combination, of a continuous function space on compact set ($$C$$) ($$\Lambda f$$) represents the integration of the function ($$\int f du$$) (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let $$\Lambda$$ be a positive linear functional on $$C_c(X)$$.

## Functional analysis

Differential equation solution is infinite function series. The infinite function series can be a sort of linear combination of countable function vector, in terms of linear algebra. This raises the problem of the analysis of function. The problem includes a distance of two functions, ie norm, completeness (Banach space). Because the series adds the last term without changing the existing terms, orthogonality is required to make a linear combination of countable functional vector becomes infinite function series (Hilbert space).

## Carathéodory's extension theorem

The studying sometimes starts with learning of boring preceding concepts. The highlight comes later. In history, the highlight concepts or the important problem were centered and the supporting concepts or lemmas followed. One of the central ideas of analysis is extension. The set of a rational number ($$\mathbb{Q}$$) extends to the real line $$\mathbb{R}$$. The Jordan measurable sets extend to the Lebesgue measurable sets ( $$\sigma -algebra$$ ). The outer measure can measure all subsets of $$X$$, whereas measure can only measure a $$\sigma -algebra$$ of measure set.