Spaces of Smooth Functions

Let $\mathcal D(\mathbb {N})$ be the space of finitely nonzero sequences (i.e., sequences with only finitely many terms different from zero). For any sequence $\alpha =(\alpha_1,\alpha_3,\ldots )$ of positive numbers let the seminorm $p_\alpha$ in $\mathcal D(\mathbb {N})$ be defined by

Let $A$ be a linear mapping of $\mathcal D(\Omega )$ to a locally convex space $L$. Prove that the following statements are equivalent:

1. $A$ is a continuous mapping;

2. $A$ is a bounded mapping (i.e., carries bounded sets into bounded sets);

3. $A$ is sequentially continuous (i.e., if $\varphi_n\to 0$ as $n\to \infty$, then $\lim_{n\to \infty }A\varphi_n=0$);

4. the restriction of $A$ to any subspace $\mathcal D_K(\Omega )\subset \mathcal D(\Omega )$ is continuous.

Prove that $\mathcal D_K(\Omega )$ is closed in $\mathcal D(\Omega )$.

Let $K$ be a compact subset of a domain $\Omega \subset \mathbb {R}^{n}$, and $\left\{ U_i\right\}$ an open covering of $K$. Prove that there exist nonnegative functions $\varphi_i\in \mathcal D(\Omega )$, $i=1,\ldots ,N$, such that:

1. $\operatorname {supp}\varphi_i\subset U_i$ for all $i$;

2. $\sum_{i=1}^N\varphi_i(x)=1$ for $x\in K$. The collection $\left\{ \varphi_i\right\}$ is called a partition of unity on $K$.

Prove that $\mathcal D(\Omega )$ is dense in $\mathcal E(\Omega )$ for any domain $\Omega \subset \mathbb {R}^{n}$.

Prove that any closed subset of $\mathbb {R}^{n}$ is the set of zeros of some function $f\in \mathcal E(\mathbb {R}^{n})$.

Let $\left\{ c_n\right\}$ be any numerical sequence. Does there exist a function $f\in \mathcal D(\mathbb {R})$ for which $f^{(n)}(0)=c_n$, $n=0,1,2,\ldots$?

Prove that the operations of differentiation $\partial /\partial x_i$ and of multiplication by the independent variable $x_i$ are continuous operators in the spaces $\mathcal D(\mathbb {R}^{n})$, $S(\mathbb {R}^{n})$, $\mathcal E(\mathbb {R}^{n})$.

Prove that if $f\in \mathcal D(\Omega )$, $g\in \mathcal E(\Omega )$, then $fg\in \mathcal D(\Omega )$. Determine whether the bilinear mapping

Let $f$ be a bounded infinitely differentiable function on the line. Determine whether multiplication by $f$ is a continuous operator

Let $G(\mathbb {R}^{2})$ be the subspace of $\mathcal E(\mathbb {R}^{2})$ consisting of the functions $\varphi$ having the property that

Prove that the operator $A$ acting according to the formula

carries $S(\mathbb {R})$ to $G(\mathbb {R}^{2})$.

The space $\mathcal D(T^n)$ of infinitely differentiable functions on the $n$-dimensional torus $T^n\simeq \mathbb {R}^{n}/\mathbb {Z}^n$ is defined as the collection of functions $\varphi$ on $T^n$ for which the corresponding functions $\Phi (t_1,\ldots ,t_n)=\varphi (e^{2\pi it_1},\ldots ,e^{2\pi it_n})$ on $\mathbb {R}^{n}$ belong to $\mathcal E(\mathbb {R}^{n})$. Prove the isomorphisms:

Let $\left\{ \delta_k\right\}$ be a sequence of positive numbers for which the series $\sum_{k=1}^\infty \delta_k$ converges. Define a sequence of functions $\left\{ f_n\right\}$ on the line by setting $f_0(x)=\operatorname {sgn}x$, $f_n(x)=(1/\delta_n)\int_{x-\delta_n}^x f_{k-1}(x)\, dx$ for $n\ge 1$. Prove that the sequence $f_n$ converges uniformly to a function $f\in \mathcal E(\mathbb {R})$ with the properties:

1. $f(x)=-1$ for $x<0$, $f(x)=+1$ for $x>\sum_{k=1}^\infty \delta_k$;

2. $\left|f^{(n)}(x)\right|\le 2^n(\delta_1\cdots \delta_n)^{-1}$ for all $x\in \mathbb {R}$.

Suppose that a countably normed space $L$ with system of seminorms $\left\{ p_k\right\}$ has the property that every set bounded with respect to $p_{k+1}$ is pre-compact with respect to $p_k$.

Let $L$ be a complete LCS and $\Omega$ a domain in $\mathbb {R}^{n}$. We define the space $\mathcal E(\Omega ,L)$ as the collection of infinitely differentiable functions on $\Omega$ with values in $L$. If $\left\{ p_\alpha \right\}_{\alpha \in A}$ is the family of seminorms that determines the topology in $L$, then the topology in $\mathcal E(\Omega ,L)$ is given by the family of seminorms

where $K$ is a compact subset of $\Omega$ and $l=(l_1,\ldots ,l_n)$ is a multi-index, with $\alpha \in A$. Prove that $\mathcal E(\Omega ,L)$ is a complete LCS, metrizable if $L$ is metrizable.

Let $\Omega_1$ and $\Omega_2$ be domains in $\mathbb {R}^{n}$ and $\mathbb {R}^{m}$. Then $\Omega_1\times \Omega_2$ is a domain in $\mathbb {R}^{n+m}$. In the notation of Problem 427 prove that

Prove that

Formulate and prove analogues of Problem 427--482 for the spaces $\mathcal D(\Omega )$ and $S(\mathbb {R}^{n})$.