Inductive Limits and the Space of Distributions

Let $d \ge 1$ and let $\Omega$ be an open subset of $\mathbb {R}^d$. Denote by $C_c^{(\infty )}(\Omega )$ all the functions $\phi : \Omega \to \mathbb {F}$ such that $\phi$ is infinitely differentiable and has compact support in $\Omega$. (The support of $\phi$ is defined by $\operatorname {spt} \phi \equiv \operatorname {cl}\left\{ x : \phi (x) \ne 0\right\}$.) If $K$ is a compact subset of $\Omega$, define $\mathscr {D}(K) \equiv \left\{ \phi \in C_c^{(\infty )}(\Omega ) : \operatorname {spt} \phi \subseteq K\right\}$. Let $\mathscr {D}(K)$ have the topology defined by the seminorms

where $k = (k_1, \dots , k_d)$, $k_j \in \mathbb {N} \cup \left\{ 0\right\}$, $\left|k\right| = k_1 + \cdots + k_d$, and $\phi^{(k)} = \partial^{\left|k\right|}\phi / \partial x_1^{k_1} \cdots \partial x_d^{k_d}$. Then $\left(C_c^{(\infty )}(\Omega ), \left\{ \mathscr {D}(K) : K \text{ is compact in } \Omega \right\} \right)$ is an inductive system (that is, $\mathscr {D}(K) \subseteq \mathscr {D}(K')$ whenever $K \subseteq K'$, each such inclusion is continuous, and $C_c^{(\infty )}(\Omega )$ is the union of the spaces $\mathscr {D}(K)$). The space $C_c^{(\infty )}(\Omega )$ is often denoted $\mathscr {D}(\Omega )$. Verify these statements.

Recall that an inductive system is a pair $(\mathscr {X}, \left\{ \mathscr {X}_i : i \in I\right\} )$, where $\mathscr {X}$ is a vector space, $\mathscr {X}_i$ is a linear manifold in $\mathscr {X}$ that has a topology $\mathscr {T}_i$ such that $(\mathscr {X}_i, \mathscr {T}_i)$ is a LCS, and, moreover: $I$ is a directed set and $\mathscr {X}_i \subseteq \mathscr {X}_j$ if $i \le j$; if $i \le j$ and $U_j \in \mathscr {T}_j$, then $U_j \cap \mathscr {X}_i \in \mathscr {T}_i$; and $\mathscr {X} = \bigcup \left\{ \mathscr {X}_i : i \in I\right\}$.

If $(\mathscr {X}, \left\{ \mathscr {X}_i, \mathscr {T}_i\right\} )$ is an inductive system, let $\mathscr {B}$ be all convex balanced sets $V$ such that $V \cap \mathscr {X}_i \in \mathscr {T}_i$ for all $i$. Let $\mathscr {T}$ be the collection of all subsets $U$ of $\mathscr {X}$ such that for every $x_0$ in $U$ there is a $V$ in $\mathscr {B}$ with $x_0 + V \subseteq U$. Then $(\mathscr {X}, \mathscr {T})$ is a (not necessarily Hausdorff) LCS. Fill in the details of the proof of this statement: specifically, show that $\mathscr {T}$ is a topology.

Prove the following. Let $(\mathscr {X}, \mathscr {T})$ be the inductive limit of an inductive system $(\mathscr {X}, \left\{ \mathscr {X}_i\right\} )$ (see Exercise 2 for the construction). Then

Let $\mathscr {X}$ be any vector space and let $\left\{ \mathscr {X}_i : i \in I\right\}$ be all of the finite dimensional subspaces of $\mathscr {X}$. Give each $\mathscr {X}_i$ the unique topology from its identification with a Euclidean space. Then $(\mathscr {X}, \left\{ \mathscr {X}_i\right\} )$ is an inductive system. Let $\mathscr {T}$ be the inductive limit topology. If $\mathscr {Y}$ is a LCS and $T : \mathscr {X} \to \mathscr {Y}$ is a linear transformation, then $T$ is $\mathscr {T}$-continuous. Verify these statements.

Let $X$ be a locally compact space and let $\left\{ K_i : i \in I\right\}$ be the collection of all compact subsets of $X$. Let $\mathscr {X}_i$ be all $f$ in $C(X)$ such that $\operatorname {spt} f \subseteq K_i$. Then $\bigcup_i \mathscr {X}_i = C_c(X)$, the continuous functions on $X$ with compact support. Topologize each $\mathscr {X}_i$ by giving it the supremum norm. Then $(C_c(X), \left\{ \mathscr {X}_i\right\} )$ is an inductive system. Verify these statements.

Let $d \ge 1$ and put $K_n = \left\{ x \in \mathbb {R}^d : \left\| x\right\| \le n\right\}$. Then $\left(\mathscr {D}(\mathbb {R}^d), \left\{ \mathscr {D}(K_n)\right\}_{n=1}^{\infty }\right)$ is an inductive system. Since a vector space together with the collection of all of its finite dimensional subspaces (each given its Euclidean topology) forms an inductive system whose inductive limit topology has the property that every linear map into a LCS is continuous for it (Example 5.9), and since the inductive limit topology is unaffected by using a cofinal subfamily of the defining subspaces to compute it (Proposition 5.8), the inductive limit topology defined on $\mathscr {D}(\mathbb {R}^d)$ by the system $\left\{ \mathscr {D}(K_n)\right\}$ equals the inductive limit topology defined by the full system $\left\{ \mathscr {D}(K) : K \text{ compact in } \mathbb {R}^d\right\}$ of Exercise 1. Verify this statement.

Let $X$ be a locally compact space, let $\left\{ K_i : i \in I\right\}$ be the collection of all compact subsets of $X$, and give $C_c(X)$ (the continuous functions on $X$ with compact support) the inductive limit topology of the system $\left\{ \mathscr {X}_i\right\}$, where $\mathscr {X}_i = \left\{ f \in C(X) : \operatorname {spt} f \subseteq K_i\right\}$ carries the supremum norm (Example 5.10). Show that if $X$ is $\sigma$-compact, then the dual of $C_c(X)$ is the space of all extended $\mathbb {F}$-valued measures.

Let $X$ be a locally compact space. As in Exercise 7, give $C_c(X)$ the inductive limit topology defined by the subspaces $\mathscr {X}_i = \left\{ f \in C(X) : \operatorname {spt} f \subseteq K_i\right\}$ (each with its supremum norm), where $\left\{ K_i\right\}$ ranges over the compact subsets of $X$. Separately, $C(X)$ (all continuous functions from $X$ into $\mathbb {F}$) carries the locally convex topology of uniform convergence on compact subsets, defined by the seminorms $p_K(f) = \sup \left\{ \left|f(x)\right| : x \in K\right\}$ for $K$ compact in $X$; restricted to the subspace $C_c(X)$, this gives a second topology on $C_c(X)$. Is the inductive limit topology on $C_c(X)$ different from this topology of uniform convergence on compact subsets of $X$?

Recall that if $\mathscr {X}$ is a TVS and $\mathscr {Y} \le \mathscr {X}$, then $\mathscr {Y}$ is topologically complemented in $\mathscr {X}$ if there is a closed linear subspace $\mathscr {Z}$ of $\mathscr {X}$ such that $\mathscr {Y} \cap \mathscr {Z} = (0)$, $\mathscr {Y} + \mathscr {Z} = \mathscr {X}$, and the map of $\mathscr {Y} \times \mathscr {Z} \to \mathscr {X}$ given by $(y,z) \mapsto y+z$ is a homeomorphism (equivalently, there is a continuous linear map $P : \mathscr {X} \to \mathscr {X}$ such that $P\mathscr {X} = \mathscr {Y}$ and $P^2 = P$).

Prove the following. If $\mathscr {X}$ is a LCS and $\mathscr {Y} \le \mathscr {X}$ such that either $\dim \mathscr {Y} < \infty$ or $\dim \mathscr {X}/\mathscr {Y} < \infty$, then $\mathscr {Y}$ is topologically complemented in $\mathscr {X}$.

Let $f$ be a Lebesgue measurable function on $\Omega$ (an open subset of $\mathbb {R}^d$, $d \ge 1$) that is locally integrable (that is, $\int_K \left|f\right| \, d\lambda < \infty$ for every compact subset $K$ of $\Omega$, where $\lambda$ is $d$-dimensional Lebesgue measure). If $L_f : \mathscr {D}(\Omega ) \to \mathbb {F}$ is defined by $L_f(\phi ) = \int f\phi \, d\lambda$, then $L_f$ is a distribution (that is, a continuous linear functional on $\mathscr {D}(\Omega )$). Verify this statement.

For the remaining exercises, $\Omega$ is always an open subset of $\mathbb {R}^d$, $d \ge 1$. If $\mu$ is a measure on $\Omega$, show that $\phi \mapsto \int \phi \, d\mu$ is a distribution [on] $\Omega$.

Let $f : \Omega \to \mathbb {F}$ be a function with continuous partial derivatives and let $L_f : \mathscr {D}(\Omega ) \to \mathbb {F}$ be the distribution $L_f(\phi ) = \int f \phi \, d\lambda$ (Example 5.20). Show that for every $\phi$ in $\mathscr {D}(\Omega )$ and $1 \le j \le d$, $L_f(\partial \phi / \partial x_j) = -L_g(\phi )$, where $g = \partial f / \partial x_j$.

Exercise 12 motivates the following definition. If $L$ is a distribution on $\Omega$, define $\partial L / \partial x_j : \mathscr {D}(\Omega ) \to \mathbb {F}$ by $\partial L / \partial x_j(\phi ) = -L(\partial \phi / \partial x_j)$ for all $\phi$ in $\mathscr {D}(\Omega )$. Show that $\partial L / \partial x_j$ is a distribution.

Using the fact that a locally integrable function $f$ on $\mathbb {R}$ defines a distribution $L_f(\phi ) = \int f\phi \, d\lambda$ (Example 5.20), and that the derivative of a distribution $L$ is the distribution $\partial L/\partial x(\phi ) = -L(\phi ')$ (Exercise 13), one is justified to talk of the derivative of any locally integrable function as a distribution. By Exercise 11 we can differentiate measures. Let $f : \mathbb {R} \to \mathbb {R}$ be the characteristic function of $[0, \infty )$ and show that its derivative as a distribution is $\delta_0$, the unit point mass at $0$. [That is, $\delta_0$ is the measure such that $\delta_0(\Delta ) = 1$ if $0 \in \Delta$ and $\delta_0(\Delta ) = 0$ if $0 \notin \Delta$.]

Let $f$ be an absolutely continuous function on $\mathbb {R}$ and show that $(L_f)' = L_{f'}$, where $L_f(\phi ) = \int f\phi \, d\lambda$ is the distribution defined by $f$ (Example 5.20) and $(L_f)'(\phi ) = -L_f(\phi ')$ is its distributional derivative (Exercise 13).

Let $f$ be a left continuous nondecreasing function on $\mathbb {R}$ and show that $(L_f)'$ is the distribution defined by the measure $\mu$ such that $\mu [a,b] = f(b) - f(a)$ for all $a < b$, where $L_f(\phi ) = \int f\phi \, d\lambda$ is the distribution defined by $f$ (Example 5.20) and $(L_f)'(\phi ) = -L_f(\phi ')$ is its distributional derivative (Exercise 13).

Let $f$ be a $C^\infty$ function on $\Omega$ and let $L$ be a distribution on $\mathscr {D}(\Omega )$. Show that $M(\phi ) \equiv L(\phi f)$, $\phi$ in $\mathscr {D}(\Omega )$, is a distribution. State and prove a product rule for finding the derivative of $M$.