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. The Understaing Analysis was give me many intuitions of analysis and encouraged me to study further. The next book was Rudin’s Functional Analysis. I realized I need to go upstream to complex Analysis, topology and measure.
During the journey of exploring the analysis, I skipped proving of theorems or solving exercises. But space between lines is coming. I’m realizing that I prove the spaces. \[\Sigma^{k}_{j=1} {\lvert}a_j-a^{n}_j{\rvert}\le\epsilon \] This holds for all finite \(k\), we even have \({\lVert} a-a_n {\rVert} _{1}\le\epsilon\). This is on the way of the proof of \(l^{1}(\mathbb{N})\) of all complex-valued sequences \(a=(a_j)^{\infty}_{j=1}\) for which the norm \({\lVert} a {\rVert} _{1}\ := \Sigma^{\infty}_{j=1} {\lvert}a_j{\rvert}\).
I could not just accept that the finite sum of each small differences \(\le \epsilon\) of \({\lvert}a_j - a^n_j{\rvert}\) holds to the infinite sum. The infinite sum is a infinte series. If the infinite series is less than or equel to zero, then it converses to the zero. If the finite sum is \(\le \epsilon\) holds every \(\mathbb{N}\), by definition the infinite sum is also \(\le \epsilon\).
It takes long time to grasp the subtle mathmatical systems. For example, a series is a number in a scalar field, a sequence is a ordered set. However the long time makes the math become familar and finally will firmly grasp the subtle concepts.
