the negative Sobolev space H^{-1}
When you push a vibrating membrane, the 'force' need not be a tidy function — you might press at a single point, or along a line, giving a load that is too singular to be an ordinary function. The right-hand side f in a PDE L u = f should be allowed to be that rough. The negative Sobolev space H^{-1} is the space of admissible right-hand sides: it is big enough to contain those singular forcings, yet still tame enough that the equation has a unique solution in H^1_0.
Formally, H^{-1}(U) is the dual space of H^1_0(U): the set of all continuous linear functionals on H^1_0, that is, all the 'measuring rules' F that eat an H^1_0 function v and return a number F(v) in a bounded, linear way. Every f in L^2 gives such a rule by integration, v maps to integral of f v, so L^2 sits inside H^1_0's dual. But the dual contains more — for instance the derivative of an L^2 function, which is generally not a function. That is the precise sense of the slogan 'H^{-1} contains one derivative less than L^2': you reach H^{-1} by differentiating L^2 functions once in the weak sense. Concretely, every element of H^{-1} can be written as f_0 + div(F) for some L^2 function f_0 and some L^2 vector field F — a function plus a divergence of square-integrable things.
Why this is the natural home for the data: the weak formulation pairs the right-hand side f against test functions v exactly through the duality F(v), so the cleanest existence theory (Lax-Milgram) asks only that f lie in H^{-1}, not in L^2. This is also why H^{-1} appears as the natural target space for the Laplacian: minus the Laplacian maps H^1_0 onto its dual H^{-1} as an isomorphism — solving minus Laplacian u = f for f in H^{-1} always has exactly one solution u in H^1_0. The negative index, k = -1, literally counts derivatives in the deficit direction, mirroring the positive Sobolev spaces.
On a domain in the plane, the Dirac delta (a point load) is not in L^2 — it is not even a function. But in two dimensions and below it is a continuous functional on H^1_0, so the delta lives in H^{-1}. That is exactly why minus Laplacian G = delta (the equation defining the Green's function) has a sensible weak solution G in H^1_0 in the plane: the singular data sits in the right dual space.
Singular loads like a point force live in H^{-1}, the natural space of right-hand sides.
H^{-1} is specifically the dual of H^1_0, not of all of H^1 — the choice matters, because the boundary behaviour built into H^1_0 is what makes minus Laplacian an isomorphism onto H^{-1}. Also, 'H^{-1} contains L^2' is a statement about a natural inclusion (via the L^2 inner product), not literal set containment; identifying L^2 with part of its own dual is the standard but slightly subtle Gelfand-triple convention.