Continuous operators on test-function spaces

Prove that the operations of differentiation and multiplication by a function $\alpha \in C^{\infty }(\mathbb {R})$ are continuous in $\mathcal{D}(\mathbb {R})$, i.e. the convergence $\varphi_{n} \xrightarrow {\mathcal{D}(\mathbb {R})} \varphi$ implies that

Prove that the differentiation operator is continuous in each of the spaces $\mathcal{D}, \mathcal{S}, \mathcal{E}$.

Let $\alpha : \mathbb {R} \rightarrow \mathbb {R}$ be an infinitely smooth function. Under what conditions on the function $\alpha$ is the operator $\varphi \longmapsto \alpha \varphi$ continuous

Prove the continuity of the integration operator

acting in the space $\mathcal{D}$ and defined on the subspace $\mathcal{D}_{0} = \left\{ \varphi \in \mathcal{D}: \int_{\mathbb {R}} \varphi (t) d t = 0\right\}$.

Let $\alpha : \mathbb {R} \rightarrow \mathbb {R}$ be a monotonically increasing infinitely differentiable function.

Let $\alpha : \mathbb {R} \rightarrow \mathbb {R}$ be a compactly supported Lebesgue-summable function. Prove the continuity of the convolution operator

in the spaces $\mathcal{D}$ and $\mathcal{E}$. Prove the continuity of this operator in the space $\mathcal{S}$ in the case when $\alpha$ is a measurable rapidly decreasing (i.e., $\sup \left\{ (1+\left|t\right|)^{n}\left|\alpha (t)\right|\right\} < \infty$ for every $n = 0,1, \ldots$) function.

Let $\mathcal{S}(\mathbb {R})$ be the Schwartz space introduced in Definition 9.18. Show that the position and momentum operators map $\mathcal{S}(\mathbb {R})$ into itself, and are self-adjoint when restricted to this domain, i.e.,

for all $f \in \mathcal{S}(\mathbb {R})$. (The Schwartz space is a convenient dense subspace of $L^{2}(\mathbb {R})$, but can be replaced in this problem by some larger subspaces of $L^{2}(\mathbb {R})$ if desired.)

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 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 $f\in \mathcal{D}'(\mathbb {R}^{n})$. Prove that the operator $S(f):\mathcal{D}(\mathbb {R}^{n})\to \mathcal{E}(\mathbb {R}^{n})$ is continuous.

Prove that every continuous operator in $S(\mathbb {R}^{n})$ that commutes with the operators $M_k$, $1\leq k\leq n$, (see Problem 580) is the operator of multiplication by a function.

Prove that every continuous operator in $S(\mathbb {R}^{n})$ that commutes with the operators $M_k$ and $D_k$, $1\leq k\leq n$, (see Problem 580) is a scalar operator.

Prove the stated assertion. If $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ in $D$, then for any $k = 1,2, \ldots$