Notice that if you have a $\kappa$-(Hilbert) dimensional space, it has a dense subset of cardinality $\kappa$ (for infinite $\kappa$) -- the finite rational combinations of orthonormal basis vectors.
On the other hand, every such space has a collection of $\kappa$ disjoint, nonempty open sets: balls of radius $\frac{1}{2}$ centered at each vector of the ON basis, so it has no dense subset of any smaller cardinality.
Therefore, it is impossible to find a homeomorphism between two Hilbert spaces of different dimensions, so it is certainly not possible to find a linear one (though, of course, there are linear isomorphisms between spaces of appropriate dimensions -- I think iff $\kappa,\lambda$ are such that $\kappa^\omega=\lambda^\omega$).