**D**(X) or C

^{∞}

_{c}(X), and the distributions therefore by

**D′**(X) or C

^{-∞}(X). Because

**D**(X) is not metrizable (unless X is compact) but only an LF space, the continuity condition is a little technical to state. (An LF space is the inductive limit of an ascending sequence of Fréchet spaces: roughly, the completeness condition is as good as that of a Fréchet space but you only have a uniformity with which to state it rather than a metric.)

- All integrable functions (with respect to a fixed measure) on X are distributions: the function f corresponds to the distribution sending φ ∈
**D**(X) to ∫_{X}fφ dx. - All regular measures on X are distributions: the measure μ corresponds to the distribution sending φ ∈
**D**to ∫_{X}φ dμ. For instance, the Dirac delta function beloved of physicists is a distribution, derived from the measure consisting of a unit point mass at 0; δ(φ) = φ(0). - There are rougher examples, however. For instance, the derivative of the delta function sends φ to -φ′(0), and this distribution is neither a function nor a measure.

Note that the word *distribution* is used by statisticians to mean something very different.

For more information, consult a textbook on partial differential equations for pure mathematicians. I like the first volume of Michael Taylor's three-volume work.