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.