There is a homology between a line segment and a convex set. It is helpful to understand the convex set. A line, a line segment, and one sideline has homology to an affine set, a convex set, and a cone. A line is \(\{y|y=\theta_1 x_1 + \theta_2 x_2, \theta_1 + \theta_2 = 1\}\) if \(\theta_1, \theta_2 \in \mathbb{R}\), a line segment is if \(\theta_1, \theta_2 > 0\) and an one side line if any \(\theta_1, \theta_2 < 0\).
Finally arrive at reproducing kernel Hilbert space. https://nzer0.github.io/reproducing-kernel-hilbert-space.html
The above post introduces RKHS in Korean. It was helpful. I had struggled to understand some concepts in RKHS. What does mean Hilbert space in terms of feature expansion? (\(f:\mathcal{X} \to \mathbb{R}\), \(f \in \mathcal{H}_K\)) It was confusing the difference between \(f\) and \(f(x)\). \(f\) means the function in Hilbert space and \(f(x)\) is evaluation.
I thought that the function can be represented by the inner product of the basis of feature space \(K(\cdot,x)\) and coefficients \(f\), and the coefficients are vectors in feature space.
Here I summarize some tools for proof of the Riesz representation theorem. They are the limit of inequality of sequence and \(\epsilon\). The Rudin’s proof of the Riesz representation theorem construct measure \(\mu\) and measurable set \(\mathfrak{M}\), then prove the \(\mu\) and \(\mathfrak{M}\) have properties. Countable additivity (not subadditivity) is an important property. The strategy of proving equality (additivity) is bidirectional inequality.
Limit of inequality of sequence gives us a tool that finite inequality makes infinite inequality.
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 \(C_c(X)\).
This is a note for Rudin’s real and complex analysis chapter 1. The key concepts are \(\sigma\)-algebra, measure (\(\mu\)) zero, and linear combination. The three concepts bring me abstract integration. The \(\sigma\)-algebra makes that countable sum and measure of complement (subtract measure) can be possible. Measure zero completes the system. linear combination integrates a measurable function.
After a measure space established, Lebesgue’s monotone convergence theorem, Fatou’s lemma, and Lebesgue’s dominant convergence theorem follow.
The compact is a property of space. In a nutshell, if space is compact, we can treat the space be a finite because space has a finite subcover. A continuous function on a compact space is uniformly continuous.
Heine-Borel theorem describes the condition of compactness of finite dimensional space. Closed and bounded But the Heine-Borel theorem does not hold in an infinite-dimensional space. We need another condition.
Previously, the compact space can be finite by taking subcover.
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).
A sequence can be defined as a function on the domain of natural number like \(1, 1/2, 1/3 ... 1/n\). This sequence approach to the 0, but never touch the 0. However, people can not take their desire to link the sequence and the 0. Because \(\infty\) is not a member of the natural number even real number, another concept is necessary to link the sequence and the 0. It is the limit.
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.
The reproducing kernel hilbert space (RKHS) was my motivation to study analysis. The hilbert space is a orthogonal normed vector space. I still do not know about the meaning of “reproducing kernal”. The RKHS appeared in the book titled An Introduction to Statisitical Learning written by Hastie.
I began to google the meaning of the spaces such as the Hilbert, Banarch. I decided to read the Understaing Analysis written by Abbott.