How to use a RBF kernel to create a "Kernel Space" using the similarity of each pair of point?
Let's take a simple example, a binary classification using only 2 dimensions with only 8 observations. Consider the first 8 rows of the dataset. It is impossible to linearly separate the data using a hyper-plane.

So, we can use a kernel transform. The books usually drop you off right here(no help). It does not tell us how to transform. Transform using what? How many dimensions? What constraints? 

We could consider the simplest transformation, f(x,y)=xy. We could take this transformation and now consider it a new dimension. So we started with (D2+Class) features and we could add 1 dim (to get D3+Class) if we chose. 

Q. Does this help?  I suggest either plotting this dataset out by hand. Using f(x,y)=xy is Not very helpful. 

So let's try another, f'(x,y)=x^2 + y^2. We started with (D2+Class) features and added another dimension. We could conceivably plot D4+C, BUT using dimensions 1,2 &amp; 4 are easier to visualize. I suggest plotting by hand (for effect) {d1,d2,d4} to your graphic or plot it using 3D software.

Now ask yourself, Is this new situation linearly separable?

As for the constraints, Do you remember LaGrange multipliers? Well if you want to use a gaussian like constraint we could use a Radial Basis Function (RBF).
Where:  

RBF: K(x,y) = exp(-gamma * (||x−y||)^2)), gamma &gt; 0


The first part is kernel 101. The LaGrangian is kernel 400. lol

dataset  

        d1  d2      C          d3         d4
| row | x | y | class |f1(x,y)=xy|f'(x,y)=x^2 + y^2|
| ---:|--:|--:| -----:|---------:|----------------:|
|   1 |  1|  0|      0|         0|                1|
|   2 |  0|  1|      0|         0|                1|
|   3 | -1|  0|      0|         0|                1|
|   4 |  0| -1|      0|         0|                1|
|   5 |  2|  0|      1|         0|                4|
|   6 |  0|  2|      1|         0|                4|
|   7 | -2|  0|      1|         0|                4|
|   8 |  0| -2|      1|         0|                4|