Why do we need to specify the domain of an unbounded operator? I am learning about the fluid dynamics and I cam across the following phrase as I was reading about the Stokes operator on Wikipedia.
"Since the Stokes operator is unbounded, we must give its domain of definition"
What does the unboundeness of an operator have to do with its domain?
 A: Almost always one studies closed operators. In fact, you rarely can do much of anything with an operator that is not closable. Closed operators that are everywhere defined on a Banach space are continuous by the closed graph theorem. So the best you can hope for is that the operator is densely-defined; that leaves you having to specify the precise domain, which usually involves boundary conditions for differential operators.
A: An old result of functional analysis tells us that a symmetric, unbounded operator acting on a Hilbert space cannot be defined on the whole space but only in a dense subspace of it. It is a direct consequence of the Hellinger-Toeplitz theorem. (see for example: Riesz-Nagy, "Functional Analysis", 1955 and also Reed-Simon, "Methods of Modern Mathematical Physics", 1975).  As a consequence of this result, the investigation of the domains of such operators has always been a delicate topic especially in problems of Mathematical Physics. 
On the other hand, if an operator is defined on the whole of the space, then its domain is trivial.  
