to define hilbert spaces how we use them in real world problem? I know all the definition of Hilbert spaces? How can we say here we have to use $L^p$ space here we have to use Hilbert space or Banach space? I do not understand why we use these spaces?
 A: All finite dimensional vector spaces $R^n$ are Hilbert space, and the existence of an inner product gives us the possibility to speak of orthogonal vectors, and many other nice properties. So our usual $3$ dimensional space is an Hilbert space, but I suppose that your question is about infinite dimensional Hilbert spaces. 
These ''space of functions'' are useful in many areas of mathematics (I don't know all possible uses, but I can mention partial differential equations), but the best exemple of their use is Quantum Mechanics. Historically the development of the mathematical theory of Hilbert spaces is parallel to the development of Quantum Mechanics in the first 20-30 years of '900.
In QM the state of a particle is represented by a vector in a (complex) Hilbert space, and the observable quantities ( energy, position, moment, etc.) are represented by linear operators on this space, and the possible values of the observable are the values in the spectrum of such operator. Since the values of a physical quantity have to be real numbers, the operators have to be Hermitian and, since for many observable the possible values can be infinite, the spectrum must contain infinitely many numbers, and this cause the Hilbert space to be infinite dimensional.
How all this exactly work is not at all simple to illustrate and can not stay in an answer. There are many resources on the net where you can search for ''mathematical foundations of quantum mechanics''.
Maybe that you think that QM is not so useful in real world problems, but think that all the electronic device that you use are, in some way, applications of quantum physics.
