Distributions as Continuous Linear Functionals

Distributions as Continuous Linear Functionals#

Definition of Distributions#

A distribution on \(\mathbb{R}^d\) is a continuous linear functional on \(\mathcal{C}_c^\infty(\mathbb{R}^d)\). The space of distributions is denoted by \(\left( \mathcal{C}_c^\infty(\mathbb{R}^d) \right)^*\).

Specifically, \(T \in \left( \mathcal{C}_c^\infty(\mathbb{R}^d) \right)^*\) if and only if:

  1. \(T\) is linear.

  2. \(T\) is continuous: For every compact \(K \subset \mathbb{R}^d\), there exists \(k \geq 0\) and \(C > 0\) such that

\[ |T(\phi)| \leq C || \phi||_{\mathcal{C}^k} \]

for all \(\phi \in \mathcal{C}_c^\infty(K)\). The smallest such integer \(k\) that works for all compact subsets is called the order of the distribution.

Remark 9

Note that the strong topology on \(\mathcal{C}_c^\infty(\mathbb{R}^d)\) defined earlier determines which linear functionals \(\lambda : \mathcal{C}_c^\infty \mapsto \mathbb{R}\) are continuous. Now the continuity of \(\lambda\) means that, if \(\left\{ f_n \right\}\) sequence that converges to \(f\) in \(\mathcal{C}_c^{\infty}\) then

\[ T(\phi_n) \to T(\phi) \ \mbox{in}\ \mathbb{R}. \]

This continuity defines which functionals qualify as distributions i.e. elements of the dual space.

Topology of the Space of Distributions#

The elements of \(\left( \mathcal{C}_c^\infty(\mathbb{R}^d) \right)^*\) are called distributions. Given a distribution \(\lambda\) we define an evaluation map as

\[ \Phi_f(\lambda) := \lambda(f) = \langle f, \lambda \rangle. \]

This evaluation map takes a distribution \(\lambda\) and outputs a real number i.e. is a linear functional on the space of distributions i.e. \(\Phi_f : \left(\mathcal{C}_c^{\infty}(\mathbb{R}^d) \right)^* \mapsto \mathbb{R}\). We can use the evaluation map to define a topology on the space of distributions. This is exactly the weak-star topology from earlier. Further, note that the evaluation functionals are exactly the images of test functions under the canonical embedding \(J : \mathcal{C}_c^{\infty) \mapsto \left( \mathcal{C}_c^{\infty} \right)^{**}\) with \(J(f) = \Phi_f\).

We can now clearly state when a sequence of distributions \(\left\{ \lambda_n \right\}\) converges to \(\lambda\) in the weak-star topology if:

\[ \langle f, \lambda_n \rangle \to \langle f, \lambda \rangle\ \mbox{for all}\ f \in \mathcal{C}_c^\infty(\mathbb{R}^d). \]

In other words, a sequence of distributions converges if their values on each test function converge.

  1. The weak-star topology is the weakest (coarsest) topology so that the evaluation map is continuous.

  2. The weak-star topology is Hausdorff meaning distinct distributions can be separated by open sets.

  3. The weak-star topology is generally not meterizable.

  4. \(\mathcal{C}_c^\infty(\mathbb{R}^d)\) is not reflexive so that the canonical embedding is not surjective.

  5. This topology is what leads to the convergence in the sense of distributions one of the weakest convergence notions in analysis, but limits remain unique.

Remark 10

Separation properties in the space of distributions is like in other dual spaces. We have the following two notions of distinguishing elements.

  1. We can use distributions to separate test functions i.e. \(\phi_1, \phi_2 \in \mathcal{C}_c^\infty(\mathbb{R}^d)\) then there exists a distribution \(\lambda\) such that \(\langle \phi_1, \lambda \rangle \neq \langle \phi_1, \lambda \rangle\). This is guaranteed by the Hahn-Banach theorem. This guarantees that the space of distributions is rich enough to distinguish between different test functions.

  2. We can use test functions to separate distributions. If \(\lambda_1, \lambda_2\) are two distinct distributions then there exists a test function \(\phi\) such that \(\langle \phi, \lambda_1 \rangle \neq \langle \phi, \lambda_2 \rangle\). This effectively says that the image of the canonical embedding is sufficient to distinguish elements in the dual space.

Examples of Distributions#

Regular Distributions from Functions#

Any locally integrable function \(f \in L_{\mathrm{loc}}^{1}(\mathbb{R}^d)\) defines a distribution \(T_f\) by:

\[ \langle T_f, \phi \rangle = \int_{\mathbb{R}^d} f(x) \phi(x)\, \mathrm{d}x. \]

Such distributions are called regular distributions. When there is no ambiguity, we often identify \(f\) with \(T_f\). Note that any absolute continuous measure with respect to the Lebesgue measure is representable by a locally integrable function. Thus we can view regular distributions as being those generated by absolutely continuous measures. If \(f\) is continuous and bounded then \(T_f\) has order 0.

Distributions from Measures#

Any Radon measure \(\mu\) on \(\mathbb{R}^d\) defines a distribution \(T_\mu\) by:

\[ \langle T_\mu, \phi \rangle = \int_{\mathbb{R}^d} \phi(x) \, \mathrm{d}\mu(x). \]

This generalizes the case of regular distributions, as \(T_f = T_{\mu_f}\) where \(\mu_f = f(x) \mathrm{d}x\). Futher, the class of Radon measures includes singular measures thus \(\mu = \mu_{\mathrm{ac}} + \mu_{\mathrm{sing}}\). Note that the singular measures include point masses, but also more complicated singular measures such as the Cantor measure.

The Dirac Delta and Point Distributions#

For any point \(x \in \mathbb{R}^d\), the Dirac delta distribution \(\delta(x_0)\) is defined by:

\[ \langle \delta_{x_0},\phi \rangle = \phi(x_0) \]

The order of the Dirac delta distribution has order 0, as

\[ |\langle \delta_{x_0}, \phi \rangle| = | \phi(x_0) | \leq || \phi ||_{\mathcal{C}^0}. \]

Note that the Dirac delta distribution is an example of a Radon measure.

Derivative of the Delta Function#

The derivative of the Dirac delta, denote \(\delta'\) is defined by

\[ \langle \delta', \phi \rangle = -\phi'(0). \]

More generally, for any multi-index \(\alpha\):

\[ \langle D^\alpha \delta, \phi \rangle = (-1)^{|\alpha|} D^\alpha \phi(0). \]

The order of the derivative of the delta function has order 1, and generally \(D^\alpha \delta\) has order \(\alpha\).

Convergence and Approximation#

Sequential Density#

Test functions are sequentially dense in the space of distributions: every distributions is the limit of a sequence of test functions in the weak-star topology.

If \(\{ \phi_n \}\) is a sequence of approximations to the identity (i.e. \(\phi_n \geq 0\), \(\int \phi_n = 1\), and supoprt shrinking to \(\{ 0 \}\)), then \(\phi_n \to \delta\) in the sense of distributions.

Convergence Properties#

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