 Very often, the space "T" is an interval of time over which we wish to solve some partial differential equation, and "μ" will be one-dimensional Lebesgue measure.The idea is to regard a function of time and space as a collection of functions of space, this collection being parametrized by time.For example, in the solution of the heat equation on a region Ω in R"n" and an interval of time [0, "T"], one seeks solutions
with time derivative
Here formula_8 denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in "L"²(Ω) that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω); formula_9 denotes the dual space of formula_8.