I was reading this article related to Hilbert spaces
I didn't get why the first function space is not Hilbert space. I mean I can define the same norm
$\|f\| =\max_{a\leqslant x\leqslant b} |f(x)|$ in the second function space as well. There also how can anyone generate the norm like that.
Also I didn't understand the part where they talk about RKHS. Any clarifications guys. I am not getting this
However, we can set the constant c (or, more generally, the value of $g(x)$ at any finite number of points) to an arbitrary real value. What this means is that a condition on the integrability of the function is not strong enough to guarantee that we can use it predictively, since prediction requires evaluating the function at a particular data value. This characteristic is what will differentiate reproducing kernel Hilbert spaces from ordinary Hilbert spaces, as we discuss in the next section.
When you change the norm of a normed vector space, you change the space - the second space with the sup-norm is a fundamentally different space than the space defined. The norm (and vector space) are both fundamental portions of a particular normed vector space - if you change either, you get a different space when talking about normed vector spaces (which inner product spaces are). In an infinite dimensional case, you can show that these norms give you different topologies. In the finite dimensional case, you know that $\mathbb{R}^n$ with the $1$ -norm has a different unit ball than $\mathbb{R}^n$ with the euclidean norm. One has an inner product ($2$-norm), the other doesn't ($1$-norm).
A norm is from an inner product if and only if it obeys parallelogram law (i.e. $2 \|x\|^2 + 2 \|y\|^2 = \|x+y\|^2 + \|x-y\|^2$). The inner product can be recovered from the norm in the case where the parallelogram law holds by the polarization identity ("the" is a bit of a misnomer - you can come up with several different polarization identities on complex vector spaces by playing with roots of unity, but they will give you the same inner product).
On proving this: It is trivial to see inner product implies parallelogram law (just use $\langle x,x \rangle = \|x\|^2$). The converse is given by polarization identity and checking properties of inner product.
See this link for more details - in the real case, the inner product the polarization identity gives you $\langle x,y \rangle = \frac{1}{4} ( \|x+y\|^2 - \|x-y\|^2)$.
The $L^2$ norm is a different norm (in fact, it is defined by the inner product, so obviously its an inner product space) than the first norm (the sup norm). Verify that the parallelogram law is violated in the sup-norm (find some examples), so it isn't even a inner product space.
A Hilbert space is a complete inner product space. You can find inner product spaces which are not complete, so after you verify that a space is or isn't an inner product space by checking the parallelogram law, you need to check if its complete (i.e. every Cauchy sequence converges) to see if the space is indeed a Hilbert space.
As for RKHS, you haven't said what you don't understand, but you can look at a book like Cucker and Zhou's "Learning Theory: An approximation viewpoint" or something for more details - there are certainly a lot of papers providing introductions to RKHS's and various note sets online like this one or this one.