Problems

Prove the stated assertion. A metric space is a Hausdorff topological space if its topology is understood as the collection of all its open sets, defined by means of the metric.

Prove the stated assertion. If $\mathcal{D}$ is a base of the topology $\Omega$ of a topological space $X$ and $\mathcal{D} \subset \mathcal{D}_{1} \subset \Omega$, then $\mathcal{D}_{1}$ is also a base of the topology of this space.

Prove the stated assertion. If $X$ is a topological space with topology $\Omega = \left\{ G\right\}$ and $E \subset X$, then the set $E$ is a topological space with the topology $\Omega_{E} = \left\{ G \cap E\right\} , G \in \Omega$. The topology $\Omega_{E}$ is called the topology induced by the topology $\Omega$ of the space $X$ on its subset $E$.

Prove the stated assertion. In a metric space, a base of its topology (see problem 1) is the collection of all $\varepsilon$-neighborhoods of all its points.

Prove the stated assertion. In a metric space, a base of its topology (see problem 1) is the collection of all $\varepsilon$-neighborhoods of all its points, where $\varepsilon$ is a rational number.

Prove the stated assertion. In a separable metric space there exists a countable base of its topology (see problem 1).

Prove the stated assertion. For any point of a metric space, its local base of the topology is formed by all $\varepsilon$-neighborhoods of this point, where $\varepsilon = 1 / n$, $n = 1,2, \ldots$

Prove the stated assertion. The union of the local bases of the topology of all points of a topological space forms a base of the topology of this space.

Prove the stated assertion. The union of two topologies on the same set may fail to be a topology on it.

Prove the stated assertion. The union of a finite collection and the intersection of any collection of closed sets is a closed set.

Prove the stated assertion. The closure of any set in a topological space is a closed set.

Prove the stated assertion. The closure of any set in a topological space is contained in every closed set containing the given set, i.e., the closure of a set is the smallest closed set containing it.

Prove the stated assertion. In a Hausdorff topological space every point is a closed set.

Give an example of a non-Hausdorff topological space in which every point is a closed set.

Prove that if $X$ is an infinite set and the topology on $X$ consists of the complements of all finite subsets of $X$, then any infinite set is dense in $X$.

Prove that in a Hausdorff topological space a sequence of points can have only one limit.

Let $X$ be the set of all functions $x:[0 ; 1] \rightarrow R$. Define neighborhoods of a point $x_{0} \in X$ as follows: choose an arbitrary $\varepsilon > 0$ and some finite set of points $t_{k} \in [0 ; 1], k = 1,2, \ldots , n$. Define the neighborhood $U\left(x_{0} ; \varepsilon ; t_{1} ; t_{2} ; \ldots ; t_{k}\right)$ of the point $x_{0}$ as the collection of all functions $x \in X$ such that for all $k = 1,2, \ldots , n$ the inequality

holds. Denote the resulting topological space by $M[0 ; 1]$. Prove that for $\lim_{n \rightarrow \infty } x_{n} = x$ to hold in this space, it is necessary and sufficient that for every point $t \in [0 ; 1]$ the equality

hold (in other words, convergence of a sequence of functions in the constructed space $M[0 ; 1]$ means pointwise convergence of this sequence).

Prove that if $X$ is a topological space, $E \subset X$, $x_{n} \in E, n = 1,2, \ldots$, and $x_{0} = \lim_{n \rightarrow \infty } x_{n}$, then $x_{0} \in \bar{E}$.

Give an example of a topological space, a subset $E$ of it, and a point $x_{0} \in \bar{E}$, for which there is no sequence $x_{n} \in E, n = 1,2, \ldots$, such that $\lim_{n \rightarrow \infty } x_{n} = x_{0}$.

Prove that in the topological space $M[0 ; 1]$ (see problem 17) the set of continuous functions $x:[0 ; 1] \rightarrow R$ is dense.

Prove that in the topological space $M[0 ; 1]$ (see problem 17) not every function of this space is the limit in it of a sequence of continuous functions.

Describe all convergent sequences of a topological space whose points are the real numbers, and whose base of the topology consists of the following sets:

Prove that the intersection of any finite collection of sets belonging to some filter is nonempty.

Verify that the set forms a filter. The $\varepsilon$-neighborhoods $\left(x_{0}-\varepsilon ; x_{0}+\varepsilon \right)$ on the real line $R$ of a given point $x_{0} \in R$.

Verify that the set forms a filter. The punctured $\varepsilon$-neighborhoods $\left(x_{0}-\varepsilon ; x_{0}\right) \cup \left(x_{0} ; x_{0}+\varepsilon \right)$ on the real line of a given point $x_{0} \in R$.

Verify that the set forms a filter. Intervals of the form $(1 ; \beta ), \beta > 1$.

Verify that the set forms a filter. The subsets $A_{n} = \left\{ n+1 ; n+2 ; \ldots \right\}$ of the set of natural numbers $N$ (this filter is called the natural filter and denoted $F_{N}$).

Verify that the set forms a complete filter ($X$ being a given set). $f = \left\{ A \mid x \in A \subset X\right\}$, where $x$ is a fixed element of the set $X$.

Verify that the set forms a complete filter ($X$ being a given set). $f = \left\{ B \mid A \subset B \subset X\right\}$, where $A$ is a fixed subset of the set $X$.

Verify that the set forms a complete filter ($X$ being a given set). The complements in the set of natural numbers $N$ of all finite subsets (this filter is called the Fréchet filter and denoted $f_{N}$).

Prove the stated assertion. The filters in problems $24-27$ are not complete.

Prove the stated assertion. The local base of the topology at any point of a topological space is a filter.

Prove the stated assertion. If $X$ is a topological space, $x$ is a limit point of it, $\mathcal{D}(x)$ is a local base of the topology at this point, and $\stackrel{\circ }{\mathcal{D}}(x)$ is the set of all punctured neighborhoods $\stackrel{\circ }{U}(x)$ of this base: $\stackrel{\circ }{U}(x) = U(x) \backslash \left\{ x\right\}$, then $\stackrel{\circ }{\mathcal{D}}(x)$ forms a filter.

Prove the stated assertion. The filters in problems 27 and 30 are equivalent.

Prove the stated assertion. If $\mathcal{D}(x)$ is a local base of the topology of a point $x$ of a metric space, consisting of all its $\varepsilon$-neighborhoods, and $\mathcal{D}_{0}(x)$ is its local base of the topology consisting only of $\varepsilon$-neighborhoods of radius $\varepsilon = 1 / n, n = 1,2, \ldots$, then the filters $\mathcal{D}(x)$ and $\mathcal{D}_{0}(x)$ are equivalent, and moreover the filter $\mathcal{D}_{0}(x)$ is a base of the filter $\mathcal{D}(x)$.

Prove the stated assertion. The natural filter $F_{N}$ (see problem 27) is a base of the Fréchet filter (see problem 30).

Prove the stated assertion. Every filter is a base of some complete filter.

Prove the stated assertion. If $f_{1}$ is a filter on a set $X_{1}, f_{2}$ is a filter on a set $X_{2}$, and $f = \left\{ C = A \times B \mid A \in f_{1}, B \in f_{2}\right\}$, then $f$ is a filter on the product $X_{1} \times X_{2}$ of the sets $X_{1}$ and $X_{2}$. (The filter $f$ is called the product of the filters $f_{1}, f_{2}$ and one writes $f = f_{1} \times f_{2}$.)

Prove the stated assertion. If $X = N$ is the set of natural numbers with the discrete topology (every point is an open set), then the natural filter $F_{N}$ (see problem 27) has no limit in $N$.

Prove the stated assertion. If $X = N \cup \left\{ +\infty \right\}$ and the local base of the topology $\mathcal{D}(+\infty )$ consists of all the sets $A_{n}$ introduced in problem 27, while the local base $\mathcal{D}(n), n \in N$, consists of the single point $n$, then the natural filter $F_{N}$ has a limit: $\lim F_{N} = +\infty$.

Prove the stated assertion. For every filter of a topological space to have at most one limit, it is necessary and sufficient that the space be Hausdorff.

Prove the stated assertion. For a point $x$ of a topological space to be the limit of some filter of this space, it is necessary that this point be the limit of every base of the filter, and sufficient that it be the limit of at least one of its bases.

Prove the stated assertion. If $X = N, Y$ is a topological space, $F_{N}$ is the natural filter (see problem 27), then the limit $\lim_{F_{N}} f(n)$ of the mapping $f(n) = y_{n} \in Y$ coincides with the limit of the sequence $\left(y_{1} ; \ldots ; y_{n} ; \ldots \right)$ in the space $Y$.

Prove the stated assertion. If $X = N \times N, Y$ is a topological space, $F_{N}$ is the natural filter (see problem 27)$, f = F_{N} \times F_{N}$ (see problem 38), $f: N \times N \rightarrow Y, f(m ; n) = y_{m n}$, then the limit $\lim_{f} f(m ; n)$ coincides with the limit of the double sequence $\lim_{(m, n) \rightarrow \infty } y_{m n}$ in the space $Y$.

Construct a set $X$, a mapping $f: X \rightarrow R$, and a filter $f$ on the set $X$ so that the limit of the Riemann integral sums of a function $\varphi : E \rightarrow R$ on a given Jordan-measurable set $E \subset R^{n}$ (i.e., the integral $\int_{E} \varphi (t) d t$) coincides with the limit of the mapping $f: X \rightarrow R$ with respect to the filter $f$.

If $X$ and $Y$ are topological spaces; $f: X \rightarrow Y$ and the filter $f$ consists of neighborhoods of some local base of the topology $\mathcal{D}\left(x_{0}\right)$ at the point $x_{0}$, then the existence of the limit $\lim_{\mathbf{f}} f(x)$ is equivalent to the continuity of the mapping $f$ at the point $x_{0}$, and moreover $\lim_{\mathbf{f}} f(x) = f\left(x_{0}\right)$.

If $X$ and $Y$ are topological spaces, $f: X \rightarrow Y$, $x_{0} \in X$, and the filter $f$ consists of the punctured neighborhoods of some local base of the topology at the point $x_{0}$, then the existence of the limit $\lim_{\mathbf{f}} f(x)$ is equivalent to the existence of the limit of the mapping $f$ over the set $X \backslash \left\{ x_{0}\right\}$, and moreover

If $X$ is an arbitrary set, $f$ is a filter on $X, Y$ a linear space, $f: X \rightarrow Y, g: X \rightarrow Y$, the mappings $f$ and $g$ have a limit with respect to the filter $f, \lambda$ and $\mu$ are numbers, then the mapping $\lambda f+\mu g$ also has a limit with respect to the filter $f$ and

Every filter on a metric space that is stronger than some local base of the topology at a point of this space is a Cauchy filter.

For a mapping $f: X \rightarrow Y$ of an arbitrary set $X$ into a complete metric space $Y$ to have a limit with respect to some filter $f$ of the set $X$, it is necessary and sufficient that the image $f(\mathbf{f})$ of the filter $f$ under the mapping $f$ be a Cauchy filter in the space $Y$.

Let $X$ and $Y$ be arbitrary sets, $Z$ a topological space, $f: X \rightarrow Y, g: Y \rightarrow Z, f_{X}$ and $f_{Y}$ filters on the spaces $X$ and $Y$ respectively, such that the filter $f\left(f_{X}\right)$ is stronger than the filter $f_{Y}$. Then, if the mapping $g$ has a limit with respect to the filter $f_{Y}$, the composition $g f$ has a limit with respect to the filter $f_{X}$ and

Prove the stated assertion. If $\varphi \in D$, then for any $k = 1,2, \ldots \quad \varphi^{(k)} \in D$.

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

Prove the stated assertion. The function $\varphi_{a}(x) = \begin{cases} e^{-a^{2} /\left(a^{2}-x^{2}\right)}, & \text{ if }\left|x\right| < a, \\ 0, & \text{ if }\left|x\right| \geqslant a, \end{cases}$ belongs to $D$.

Prove the stated assertion. For a function $\varphi \in D$ there exists a function $\psi \in D$ such that $\varphi = \psi^{\prime }$ if and only if $\int_{-\infty }^{+\infty } \varphi (x) d x = 0$.

Prove the stated assertion. Two functions continuous on the real line are distinct if and only if the generalized functions generated by them are distinct.

Prove the stated assertion. The functional v.p. $\int_{-\infty }^{+\infty } \frac{\varphi (x)}{x} d x, \quad \varphi \in D$, is a generalized function (it is usually denoted $\mathcal{P} \frac{1}{x}$).

Prove the stated assertion. The $\delta$-function is not generated by any locally integrable function.

Prove the stated assertion. The $\delta$-function is the limit in $D^{\prime }$ of a sequence of generalized functions generated by locally integrable functions.

Do the limits $\lim_{n \rightarrow \infty } \cos n x$, $\lim_{n \rightarrow \infty } \sin n x$ exist in the space $D^{\prime }$? If they exist, what are they equal to?

If $f_{n} \in D^{\prime }, n = 1,2, \ldots$, and for every function $\varphi \in D$ the limit of the numerical sequence $\left(f_{n}, \varphi \right)$ exists, then the functional $F$ defined by the equality $(F, \varphi ) = \lim_{n \rightarrow \infty }\left(f_{n}, \varphi \right)$ is a generalized function: $F \in D^{\prime }$.

If a sequence of absolutely integrable functions $f_{n}(l), n = 1,2, \ldots$, is such that:

1. whatever the number $c > 0$ may be, for $\left|a\right| < c,\left|b\right| < c$ the sequence

   $$\left|\int _{a}^{b} f_{n}(t) d t\right|, \quad n = 1,2, \ldots$$

   is bounded above;

2. for any fixed $a$ and $b$, different from zero,

   $$\lim _{n \rightarrow \infty } \int _{a}^{b} f_{n}(x) d x = \begin{cases} 0 & \text{ for } a < b < 0 \text{ and } 0 < a < b, \\ 1 & \text{ for } a < 0 < b \end{cases}$$

then it is called $\delta$-shaped.

For every continuous function $\varphi : R \rightarrow R$ and every $\delta$-shaped sequence $\left\{ f_{n}\right\}$ the equality

holds.

If $f_{t}(x) = \frac{1}{\sqrt{\pi t}} e^{-x^{2} / t}$, then in the space $D^{\prime }$ the equality $\lim_{t \rightarrow +0} f_{t}(x) = \delta (x)$ holds.

In the space $D^{\prime }$ there exists the limit $\lim_{y \rightarrow +0} \frac{1}{x \pm i y}$ (it is denoted $\frac{1}{x \pm i 0}$), and the formula

holds (see problem 57).

Every generalized function is the limit of generalized functions generated by locally integrable functions. In this sense the space of generalized functions is a "completion" of the space of ordinary locally integrable functions.

A generalized function $f \in D^{\prime }$ is said to vanish on the interval $(a ; b)$ if for all $\varphi \in D, \operatorname {supp} \varphi \subset (a ; b)$, the equality $(f ; \varphi ) = 0$ holds. For a continuous function to vanish at every point of the interval, it is necessary and sufficient that it vanish on this interval as a generalized function.

If a function $f: R \rightarrow C$ is continuously differentiable, then for the generalized function generated by its derivative $f^{\prime }$ the relation $\left(f^{\prime }, \varphi \right) = -\left(f, \varphi^{\prime }\right), \quad \varphi \in D$, holds.

The derivative $f^{\prime }$ of a generalized function $f \in D^{\prime }$ is also a generalized function in $D^{\prime }$.

The derivative of any order of a generalized function from the space $D^{\prime }$ is a generalized function in the same space.

If $f$ and $g$ are generalized functions, and $\lambda , \mu \in C$, then $(\lambda f + \mu g)^{\prime } = \lambda f^{\prime }+\mu g^{\prime }$.

Compute the derivative of the generalized function. $y = \theta \left(x-x_{0}\right) = \begin{cases} 1 & \text{ for } x \geqslant x_{0}, \\ 0 & \text{ for } x < x_{0}. \end{cases}$

Compute the derivative of the generalized function. $y = \delta \left(x-x_{0}\right)$.

Compute the derivative of the generalized function. $y = \left|x\right|$.

Compute the derivative of the generalized function. $y = x_{+}$, where $x_{+} = \begin{cases} x & \text{ for } & x \geqslant 0, \\ 0 & \text{ for } & x < 0. \end{cases}$

Compute the derivative of the generalized function. $y = \ln \left|x\right|$.

Compute the derivative of the generalized function. $y = x_{+}^{\lambda } = \begin{cases} x^{\lambda } & \text{ for } x > 0, \\ 0 & \text{ for } x < 0, \end{cases} \quad -1 < \lambda < 0$.

Compute the derivative of the generalized function. $y = \ln x_{+} = \begin{cases} \ln x & \text{ for } & x > 0, \\ 0 & \text{ for } & x < 0. \end{cases}$

Find the derivatives of order $n$ of the generalized function. $y = \delta (x)$.

Find the derivatives of order $n$ of the generalized function. $y = x_{+}^{k}, k = 0,1,2, \ldots$

Prove the stated assertion. $\left(\frac{d}{d x}+\lambda \right) \theta (x) e^{-\lambda x} = \delta (x)$.

Prove the stated assertion. $\left(\frac{d^{2}}{d x^{2}}+\omega^{2}\right) \frac{\theta (x) \sin \omega x}{\omega } = \delta (x)$.

Prove the stated assertion. If $\delta_{\varepsilon }(x) = \begin{cases} 1 / \varepsilon & \text{ for } & \left|x\right| < \varepsilon / 2, \\ 0 & \text{ for } & \left|x\right| \geqslant \varepsilon / 2, \end{cases}$ then in the space $D^{\prime }$ the limit

exists.

Prove the stated assertion. If

where the functions $f_{1}$ and $f_{2}$ are continuous and piecewise continuously differentiable on $R$ (so that, in particular, the limits $f(x_{0} \pm 0)$ exist), and

(see problem 71), then the function $g(x)$ is continuous on $R$, has a locally integrable derivative $g^{\prime }$, and

Prove the stated assertion. If $f$ is a piecewise smooth function on $R$, having at the points $x_{1}, \ldots , x_{n}$ jump discontinuities of the first kind with jumps $p_{1}, p_{2}, \ldots , p_{n}$, then

where $f^{\prime }$ is the generalized derivative of the function $f$, and $\frac{d f(x)}{d x}$ is the generalized function generated by the ordinary derivative of the function $f$ for $x \neq x_{k}, k = 1,2, \ldots , n$.

Prove the stated assertion. If $f_{n} \in D^{\prime }, n = 1,2, \ldots , f \in D^{\prime }$ and $\lim_{n \rightarrow \infty } f_{n} = f$, then for any $k = 1,2, \ldots$ in the space $D^{\prime }$ the equality $\lim_{n \rightarrow \infty } f_{n}^{(k)} = f^{(k)}$ holds.

Prove the stated assertion. In the space of generalized functions, convergent series can be differentiated term by term any number of times: if $f = \sum_{n = 1}^{\infty } f_{n}$, then $f^{(k)} = \sum_{n = 1}^{\infty } f_{n}^{(k)}$ in $D^{\prime }, k = 1,2, \ldots$

Prove the stated assertion. In the space $D^{\prime }$ the equalities

hold.

Prove the stated assertion. For an infinitely differentiable function $\varphi$ : $R \rightarrow C$ to belong to the space $S$, it is necessary and sufficient that for any nonnegative integers $m$ and $n$ the condition

hold.

Prove the stated assertion. For $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ to hold, it is necessary and sufficient that

Prove the stated assertion. If $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ in $S$, then for any $m = 1,2, \ldots$ and any $x \in R$

holds.

Prove the stated assertion. The space of test functions $D$ is contained in the space of test functions of slow growth $S$, and if $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ in $D$, then $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ in $S$.

Prove the stated assertion. $D \neq S$.

Prove the stated assertion. The generalized function generated by the locally integrable function $e^{x}$ cannot be extended from the set of test functions $D$ to the set of test functions $S$, i.e., it cannot be extended to an element of the space $S^{\prime }$.

Prove the stated assertion. The space $D$ is dense in the space $S$, i.e., any function $\varphi \in S$ is the limit in $S$ of a sequence of functions $\varphi_{n} \in D, n = 1,2, \ldots$

Prove the stated assertion. If a function $f(x)$ is absolutely integrable over the whole real line, then the functional $f$ defined by the formula

belongs to the space $S^{\prime }$.

Prove the stated assertion. If a function $f: R \rightarrow C$ is locally integrable and satisfies the estimate $\left|f(x)\right| \leqslant c\left|x\right|^{k}$, where $x$ and $k$ are nonnegative constants, then the functional $f$ defined by the formula

belongs to the space $S^{\prime }$.

Prove the stated assertion. The generalized function $\frac{1}{x+i 0} \in D^{\prime }$ (see problem 64) can be extended to a generalized function of slow growth.

Prove the stated assertion. For derivatives of generalized functions of slow growth, complete analogues of the assertions of problems $69, 70, 85$ and 86 hold.

Prove the stated assertion. If $\varphi \in S$, then for any $k = 1,2, \ldots$ the function $x^{k} \varphi (x)$ is absolutely integrable over the whole real line, and hence its Fourier transform is defined.

Prove the stated assertion. If $\varphi \in S$, then $F(\varphi ) \in S$ and $F^{-1}(\varphi ) \in S$ ($F$ being the Fourier transform, and $F^{-1}$ the inverse Fourier transform).

Prove the stated assertion. If $\varphi \in S$, then $F^{-1}(F(\varphi )) = \varphi$ and $F\left(F^{-1}(f)\right) = \varphi$.

Prove the stated assertion. The Fourier transform and the inverse Fourier transform map the space $S$ onto itself one-to-one, linearly, and continuously. (A mapping $F: S \rightarrow S$ is called continuous if the condition $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi , \varphi_{n} \in S, n = 1,2, \ldots$, implies that $\lim_{n \rightarrow \infty } F\left(\varphi_{n}\right) = F(\varphi )$.)

Prove the stated assertion. If a function $f(x)$ is continuous and absolutely integrable over the whole real line and $\varphi \in S$, then

Give an example of a test function $\varphi$ from the space $D$ such that its Fourier transform $F(\varphi )$ does not belong to this space.

Find the direct and inverse Fourier transforms for the $\delta$-function.

Prove the stated assertion. The Fourier transform of a generalized function $f \in S^{\prime }$ is also a generalized function of class $S^{\prime }$, i.e., $F(f)$ is a linear continuous functional on the space $S$.

Prove the stated assertion. For every generalized function $f \in S^{\prime }$ the relations $F^{-1}(F(f)) = F\left(F^{-1}(f)\right) = f$ hold.

Prove the stated assertion. The direct $F$ and inverse $F^{-1}$ Fourier transforms map the space $S^{\prime }$ onto itself linearly, one-to-one, and continuously (continuity of a mapping of the space $S^{\prime }$ is defined analogously to the continuity of the mapping $S \rightarrow S$ in problem 102).

Prove the stated assertion. Let $\psi : R \rightarrow C$ be an infinitely differentiable function such that for every derivative $\psi^{(n)}, n = 0,1,2, \ldots$, there exist constants $C_{n}$ and $C_{n}^{\prime }$ such that for all $x \in R$ the inequality $\left|\psi^{(n)}(x)\right| \leqslant C_{n}^{\prime }(1+\left|x\right|)^{C_{n}}$ holds. Then, if $\varphi \in S$, $\psi \varphi \in S$.

Prove the stated assertion. If $f \in S^{\prime }$, and the function $\psi$ satisfies the conditions of problem 109, then $\psi f \in S^{\prime }$.

Prove the stated assertion. If $f \in S^{\prime }$, then $F\left(f^{(n)}\right) = (i x)^{n} F(f)$.

Prove the stated assertion. If $f \in S^{\prime }$, then $F^{(n)}(f) = (-i)^{n} F\left(x^{n} f\right)$.

Find the Fourier transform of the generalized function. $\delta^{(n)}$.

Find the Fourier transform of the generalized function. $\sum_{k = 0}^{n} a_{k} x^{k}$.

Find the Fourier transform of the generalized function. The shifted $\delta$-function $\delta (x-a)$.

Find the Fourier transform of the generalized function. $\frac{\delta (x-a)+\delta (x+a)}{2}$.

Find the Fourier transform of the generalized function. $\frac{\delta (x-a)-\delta (x+a)}{2}$.

Find the Fourier transform of the generalized function. The Heaviside function $\theta$ (see example 3).

Example 3 defines the Heaviside function as

a locally integrable function regarded as a generalized function; there it is shown that $\theta^{\prime } = \delta$.