Inner product in Sobolev spaces Many books state the following: 

Let $u, v \in W^{1,2}(\Omega)$ (a Sobolev space), the scalar product is:

$$u \cdot v = \int_{\Omega} uv \; dx + \int_{\Omega} \nabla u \nabla v \; dx $$ 

with $x = (x_1,x_2)$

Is that a definition or it is derived from any relationship? If is a relationship how to find it?

Is that also an inner product?
An answer I've heard follows:

One way (there are others) to analyze solutions in modern PDE analysis is to:


Specify a space of functions with certain regularity (e.g. all functions on a bounded domain that have continuous derivatives up to second order)
Imagine a PDE as an operator on this function space.


We then use tools from functional analysis to say something about whether a solution exists (maybe play some game about examining how sequences in the domain get mapped to the range under the operator defined in 2).

We need to be careful about the spaces and operators we define if we want to play this game though. Consider our first guess for how we might want to play:


Consider continuous functions on a bounded interval under the sup norm
Consider the derivative operator


If we consider the sequence $f_n(x)=\sin(nx)$, The derivative operator will give the following bound:
$$f'_n(x) = n \cos(nx) \\ ||f'_n|| \leq n ||f_n||$$
Which is to say, the derivative operator is not a continuous operator (I skipped a step, but stated the punchline). Continuous operators over spaces are things we need in order to play our game though -- think about what they do: if we have a sequence in the domain, a continuous operator allows us to say something about how the limiting object in the domain passes to the range.

So we'd like to manufacture complete spaces of functions over which we can define continuous maps. Sobolev spaces $W^{k,p}$ are one technology we have for this. What we do: control (via $L^p$ norm) how badly the first $k$ derivatives of a function can behave. These norms are created so that we can create continuous operators over these spaces.

That $W^{1,2}$ is Hilbert is coincidental, in general these things are Banach spaces.
An answer I've heard follows:

One way (there are others) to analyze solutions in modern PDE analysis is to:


Specify a space of functions with certain regularity (e.g. all functions on a bounded domain that have continuous derivatives up to second order)
Imagine a PDE as an operator on this function space.


We then use tools from functional analysis to say something about whether a solution exists (maybe play some game about examining how sequences in the domain get mapped to the range under the operator defined in 2).

We need to be careful about the spaces and operators we define if we want to play this game though. Consider our first guess for how we might want to play:


Consider continuous functions on a bounded interval under the sup norm
Consider the derivative operator


If we consider the sequence $f_n(x)=\sin(nx)$, The derivative operator will give the following bound:
$$f'_n(x) = n \cos(nx) \\ ||f'_n|| \leq n ||f_n||$$
Which is to say, the derivative operator is not a continuous operator (I skipped a step, but stated the punchline). Continuous operators over spaces are things we need in order to play our game though -- think about what they do: if we have a sequence in the domain, a continuous operator allows us to say something about how the limiting object in the domain passes to the range.

So we'd like to manufacture complete spaces of functions over which we can define continuous maps. Sobolev spaces $W^{k,p}$ are one technology we have for this. What we do: control (via $L^p$ norm) how badly the first $k$ derivatives of a function can behave. These norms are created so that we can create continuous operators over these spaces.

That $W^{1,2}$ is Hilbert is coincidental, in general these things are Banach spaces.