In the following, real-valued distributions on an
open subset U of
Rn will be formally defined. With minor modifications, one can also define complex-valued distributions, and one can replace
Rn by any (
paracompact)
smooth manifold. The first object to define is the space D(
U) of test functions on
U. Once this is defined, it is then necessary to equip it with a
topology by defining the
limit of a sequence of elements of D(
U). The space of distributions will then be given as the space of
continuous linear functionals on D(
U).
Test function space The space D(
U) of
test functions on
U is defined as follows. A function \varphi :
U →
R is said to have
compact support if there exists a
compact subset
K of
U such that \varphi(
x) = 0 for all
x in
U \
K. The elements of D(
U) are the infinitely differentiable functions \varphi :
U →
R with compact support – also known as
bump functions. This is a real
vector space. It can be given a
topology by defining the
limit of a sequence of elements of D(
U). A sequence (\varphi
k) in D(
U) is said to converge to \varphi ∈ D(
U) if the following two conditions hold: • There is a compact set
K ⊂
U containing the supports of all \varphi
k: ::\bigcup\nolimits_k \operatorname{supp}(\varphi_k)\subset K. • For each
multi-index α, the sequence of partial derivatives \partial^\alpha \varphi_k tends
uniformly to \partial^\alpha\varphi. With this definition, D(
U) becomes a
complete locally convex topological vector space satisfying the
Heine–Borel property. This topology can be placed in the context of the following general construction: let :X = \bigcup\nolimits_i X_i be a countable increasing union of locally convex topological vector spaces and ι
i :
Xi →
X be the inclusion maps. In this context, the
inductive limit topology, or
final topology, τ on
X is the finest locally convex vector space topology making all the inclusion maps \iota_i continuous. The topology τ can be explicitly described as follows: let
β be the collection of convex balanced subsets
W of
X such that
W ∩
Xi is open for all
i. A base for the inductive limit topology τ then consists of the sets of the form
x +
W, where
x in
X and
W in
β. The proof that τ is a vector space topology makes use of the assumption that each
Xi is locally convex. By construction,
β is a local base for
τ. That any locally convex vector space topology on
X must necessarily contain
τ means it is the weakest one. One can also show that, for each
i, the subspace topology
Xi inherits from τ coincides with its original topology. When each
Xi is a
Fréchet space, (
X, τ) is called an
LF space. Now let
U be the union of
Ui where {
Ui} is a countable nested family of open subsets of
U with compact closures
Ki =
i. Then we have the countable increasing union :\mathrm{D}(U) = \bigcup\nolimits_i \mathrm{D}_{K_i} where D
Ki is the set of all smooth functions on
U with support lying in
Ki. On each D
Ki, consider the topology given by the seminorms :\| \varphi \|_\alpha = \max_{x \in K_i} \left |\partial^\alpha \varphi \right |, i.e. the topology of uniform convergence of derivatives of arbitrary order. This makes each D
Ki a
Fréchet space. The resulting
LF space structure on D(
U) is the topology described in the beginning of the section. On D(
U), one can also consider the topology given by the
seminorms :\| \varphi \|_{\alpha, K_i} = \max_{x \in K_i} \left |\partial^\alpha \varphi \right | . However, this topology has the disadvantage of not being complete. On the other hand, because of the particular features of D
Ki's, a set this bounded with respect to τ if and only if it lies in some D
Ki's. The completeness of (
D(
U), τ) then follow from that of D
Ki's. The topology
τ is not
metrizable by the
Baire category theorem, since D(
U) is the union of subspaces of the
first category in D(
U).
Distributions A
distribution on
U is a
continuous linear functional T : D(
U) →
R (or
T : D(
U) →
C). That is, a distribution
T assigns to each test function \varphi a real (or complex) scalar
T(\varphi) such that : T(c_1\varphi_1 + c_2\varphi_2) = c_1 T(\varphi_1) + c_2 T(\varphi_2) for all test functions \varphi1, \varphi2 and scalars c1, c2. Moreover,
T is continuous if and only if :\lim_{k\to\infty}T(\varphi_k)= T\Bigl(\lim_{k\to\infty}\varphi_k\Bigr) for every convergent sequence \varphi
k in D(
U). (Even though the topology of D(
U) is not metrizable, a linear functional on D(
U) is continuous if and only if it is sequentially continuous.) Equivalently,
T is continuous if and only if for every compact subset
K of
U there exists a positive constant
CK and a non-negative integer
NK such that : |T(\varphi)| \le C_K \sup \bigl\{ |\partial^\alpha\varphi(x)| \mathrel{\big|} x\in K, |\alpha|\leq N_K \bigr\} for all test functions \varphi with support contained in
K. The space of distributions on
U is denoted by D′(
U) and it is the
continuous dual space of D(
U). No matter what dual topology is placed on D′(
U), a
sequence of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a
net), which is why the topology is sometimes defined to be the
weak-* topology. But often the
topology of bounded convergence, which in this case is the same as the topology of uniform convergence on compact sets, is placed on D′(
U) since it is with this topology that D′(
U) becomes a
nuclear Montel space and it is with this topology that the
kernels theorem of Schwartz holds. No matter which topology is chosen, D′(
U) will be a non-metrizable,
locally convex topological vector space. The duality pairing between a distribution
T in D′(
U) and a test function \varphi in D(
U) is denoted using
angle brackets by :\begin{cases} \mathrm{D}'(U) \times \mathrm{D}(U) \to \mathbf{R} \\ (T, \varphi) \mapsto \langle T, \varphi \rangle, \end{cases} so that
T,\varphi =
T(\varphi). One interprets this notation as the distribution
T acting on the test function \varphi to give a scalar, or symmetrically as the test function \varphi acting on the distribution
T. A sequence of distributions (
Tk) converges with respect to the weak-* topology on D′(
U) to a distribution
T if and only if :\langle T_k, \varphi\rangle \to \langle T, \varphi\rangle for every test function \varphi in D(
U). For example, if
fk :
R →
R is the function : f_k(x) = \begin{cases} k & \text{if}\ 0\le x \le 1/k \\ 0 & \text{otherwise} \end{cases} and
Tk is the distribution corresponding to
fk, then : \langle T_k, \varphi\rangle = k\int_0^{1/k} \varphi(x)\, dx \to \varphi(0) = \langle \delta, \varphi\rangle as
k → ∞, so
Tk →
δ in D′(
R). Thus, for large
k, the function
fk can be regarded as an approximation of the Dirac delta distribution.
Functions as distributions The function
f :
U →
R is called
locally integrable if it is
Lebesgue integrable over every compact subset
K of
U. This is a large class of functions which includes all continuous functions and all
Lp functions. The topology on D(
U) is defined in such a fashion that any locally integrable function
f yields a continuous linear functional on D(
U) – that is, an element of D′(
U) – denoted here by
Tf, whose value on the test function \varphi is given by the Lebesgue integral: :\langle T_f,\varphi \rangle = \int_U f\varphi\,dx. Conventionally, one
abuses notation by identifying
Tf with
f, provided no confusion can arise, and thus the pairing between
Tf and \varphi is often written :\langle f, \varphi\rangle = \langle T_f,\varphi\rangle. If
f and
g are two locally integrable functions, then the associated distributions
Tf and
Tg are equal to the same element of D′(
U) if and only if
f and
g are equal
almost everywhere (see, for instance, ). In a similar manner, every
Radon measure μ on
U defines an element of D′(
U) whose value on the test function \varphi is ∫\varphi
dμ. As above, it is conventional to abuse notation and write the pairing between a Radon measure
μ and a test function \varphi as \langle \mu, \varphi \rangle. Conversely, as shown in a theorem by Schwartz (similar to the
Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure. The test functions are themselves locally integrable, and so define distributions. As such they are
dense in D′(
U) with respect to the topology on D′(
U) in the sense that for any distribution
T ∈ D′(
U), there is a net \varphi
i ∈ D(
U) such that :\langle\varphi_i,\psi\rangle\to \langle T,\psi\rangle for all
Ψ ∈ D(
U). This fact follows from the
Hahn–Banach theorem, since the dual of D′(
U) with its weak-* topology is the space D(
U). A stronger result of sequential density can be proven more constructively by a convolution argument. == Operations on distributions ==