Understanding positive definite kernel From Mercer's Theorem:

A kernel is a symmetric continuous function $ K: [a,b] \times [a,b] \rightarrow \mathbb{R}$, so that $K(x, s) = K(s, x)$ ($\forall s,x \in [a,b]$).
$K$ is said to be non-negative definite (or positive semi-definite) if
  and only if $$\sum_{i=1}^n\sum_{j=1}^n K(x_i, x_j) c_i c_j \geq 0$$
  for all finite sequences of points $x_1, ..., x_n$ of $[a, b]$ and all
  choices of real numbers $c_1, ..., c_n$.

From Positive-definite kernel:

Let $\{ H_n \}_{n \in {\mathbb Z}}$ be a sequence of (complex) Hilbert
  spaces and $\mathcal{L}(H_i, H_j)$ be the bounded operators from $H_i$
  to $H_j$.
A map $A$ on ${\mathbb Z} \times {\mathbb Z}$ where
  $A(i,j)\in\mathcal{L}(H_i, H_j)$ is called a positive definite kernel
  if for all $m > 0$ and $h_i \in H_i$, the following positivity
  condition holds: $$\sum_{-m \leq i\quad\, \atop j \leq m} \langle
 A(i,j) h_i, h_j \rangle \geq 0. $$




*

*I wonder if the two definitions for positive-definite kernel agree
with each other and how?

*Is a positive-definite kernel related to a positive-definite bilinear
form on a vector space?

*What is the definition of kernel in its most general case, i.e. for the general Hilbert space
case?


References are also appreciated.
Thanks and regards!
 A: Those definitions are very closely related, in that they can be put into the same general framework.  
Let $X$ be a set, let $(H_x)_{x\in X}$ be a family of Hilbert spaces indexed by $X$, and for each $(x,y)\in X\times X$, let $K(x,y)$ be an element of $\mathcal{L}(H_x,H_y)$.  Then $K$ is called a positive (semidefinite) kernel if for all finite sequences $x_1,\ldots x_n$ in $X$ and $h_1,\ldots,h_n$ with $h_i\in H_{x_i}$,
$$\sum_{i,j=1}^n\langle K(x_i,x_j)h_i,h_j\rangle\geq 0.$$  This is equivalent to requiring that the matrix $(K(x_j,x_i))_{ij}$ represents a positive operator on the Hilbert space $H_{x_1}\oplus\cdots\oplus H_{x_n}$.  (There's a distracting transpose there, but I'm trying to make this consistent with the definition you gave.)
You get your second example by taking $X=\mathbb Z$.  You get your first example by taking $X=[a,b]$ and $H_x=\mathbb R$ (as a real Hilbert space) for all $x$.
It is common (for example in the context mentioned below) that there is a single Hilbert space $H$ such that $H_x=H$ for all $x\in X$.  In that case, $K:X\times X\to \mathcal{L}(H)$ is a positive kernel if for each finite sequence $x_1,\ldots,x_n$ in $X$, the matrix $(K(x_j,x_i))_{ij}$ represents a positive operator on the Hilbert space $H^{(n)}$. 
So far I hope I have somewhat answered questions 1 and 3.  Now I will discuss one context where such functions arise, leading to a partial answer to question 2.

One place where positive kernel functions arise is in the study of reproducing kernel Hilbert spaces.  If $X$ is a set, $H$ is a Hilbert space, and $E$ is a Hilbert space whose elements are $H$-valued functions on $X$, then $E$ is called a (vector-valued) reproducing kernel Hilbert space if for each $x\in X$, evaluation at $x$ is a bounded linear operator from $E$ to $H$.  
Suppose that $E$ satisfies this definition, and for each $x\in X$, let $\mathrm{ev}_x\in\mathcal{L}(E,H)$ be evaluation at $x$, $\mathrm{ev}_x(f)=f(x)$.  Let $K:X\times X\to \mathcal{L}(H)$ be defined by $K(x,y)=\mathrm{ev}_x\mathrm{ev}_y^*$.  Then $K$ is a positive kernel on $X$, called the reproducing kernel for $E$.  The reason for the name "reproducing kernel" is that the evaluations of elements of $E$ can be "reproduced" from $K$ and the inner product on $E$.  For each $x\in X$ and $h\in H$, the function $k_{x,h}:X\to H$ defined by $k_{x,h}(y)=K(y,x)h$ is in $E$.  If $f$ is in $E$, then $\langle f(x),h\rangle=\langle f,k_{x,h}\rangle$.  (In fact, note that $k_{x,h}=\mathrm{ev}_x^*h$.)  This property uniquely characterizes $K$.
Conversely, if $X$ is a set, $H$ is a Hilbert space, and $K:X\times X\to \mathcal{L}(H)$ is a positive kernel function, then there is a unique reproducing kernel Hilbert space $E_K$ of $H$-valued functions on $X$ such that for all $f\in E$, $x\in X$, and $h\in H$, $k_{x,h}$ as defined above is in $E_K$, and $\langle f(x),h\rangle=\langle f,k_{x,h}\rangle$.  In other words, $K$ is the (unique) reproducing kernel for a (unique) reproducing kernel Hilbert space.  For the construction of $E_K$, a positive bilinear (or sesquilinear) form, i.e. an inner product, is defined on a free vector space whose formal generators "wind up" being the functions $k_{x,h}$ after completion. The positivity of $K$ is precisely what is needed to make the inner product work, so this might be a partial answer to your question 2.  (I am leaving this part vague for now, but if you're interested I can elaborate or provide a reference.)

There are further generalizations that have appeared in operator theory and operator algebras literature, and there are certainly more types and applications of positive kernel functions than I am even aware of, let alone mentioned here.  I may add more references at some point (I certainly will if you ask), but for now I will point out a couple that I have found particularly useful:


*

*Aronszajn's paper "Theory of reproducing kernels" contains the first systematic study of (scalar-valued) positive kernel functions and reproducing kernel Hilbert spaces.  

*Agler and McCarthy's book Pick interpolation and Hilbert function spaces covers a number of topics related to positive kernel functions and reproducing kernel Hilbert spaces, particularly complex analytic interpolation problems and operator theory.


You might also be interested in a couple of past questions, here and here, which were about scalar-valued reproducing kernel Hilbert spaces (and hence, at least implicitly, involved positive scalar-valued kernels). 
