Function Spaces and Generalized Functions

Prove that for any measurable functions on a set $X$ with measure $\mu$ the Hölder inequality is valid:

where the numbers $p$ and $q$ are connected by the relation $1/p+1/q=1$.

Prove that

for any measurable function $f$ on a set $X$ with measure $\mu$, where the supremum is taken over all functions $g(x)$ satisfying the inequality $\int_X\left|g(x)\right|^q\, d\mu (x)\leq 1$, and the numbers $p$ and $q$ are connected by the relation $1/p+1/q=1$.

Prove the Minkowski integral inequality

for $1<p<\infty$.

Say that a measure $\mu$ on a set $X$ has a countable base if there exists a countable family $\left\{ A_n\right\}$ of measurable subsets of $X$ such that for any measurable subset $B$ there is a set $A_n$ for which $\mu (A_n\mathbin {\triangle }B)<\varepsilon$.

Prove that $L^{1}\left(X,\mu \right)$ is a separable space if and only if $\mu$ has a countable base.

Prove that $L^{p}\left(X,\mu \right)$, $1<p<\infty$, is a separable space if and only if $L^{1}\left(X,\mu \right)$ is.

Prove that $L^{\infty }\left(X,\mu \right)$ is either finite-dimensional or nonseparable.

Let $\mu (X)<\infty$. Prove that for $p\geq q\geq 1$ the space $L^{p}\left(X,\mu \right)$ is contained in the space $L^{q}\left(X,\mu \right)$.

Prove that if $p\neq q$, then neither of the spaces $L^{p}\left(\mathbb {R},dx\right)$, $L^{q}\left(\mathbb {R},dx\right)$ is contained in the other.

Let $0<\alpha \leq \beta <\infty$. For what $p$ does the function $f(x)=1/(x^\alpha +x^\beta )$ belong to $L^{p}\left(\mathbb {R}_+,dx\right)$ ($\mathbb {R}_+$ denotes the positive half-line)?

Suppose that the numbers $p$, $q$, $r$ are connected by the relation $1/p+1/q+1/r=1$, and that $f\inL^{p}\left(X,\mu \right)$, $g\inL^{q}\left(X,\mu \right)$, $h\inL^{r}\left(X,\mu \right)$. Prove that $fgh$ is an integrable function and that $\left\| fgh\right\|_{1}\leq \left\| f\right\|_{p}\left\| g\right\|_{q}\left\| h\right\|_{r}$.

Let $1\leq p\leq r\leq q\leq \infty$. Prove that $L^{p}\left(X,\mu \right)\capL^{q}\left(X,\mu \right)$ is contained in $L^{r}\left(X,\mu \right)$, and that $\left\| f\right\|_{r}\leq \left\| f\right\|_{p}^\alpha \left\| f\right\|_{q}^\beta$ for every function $f\inL^{p}\left(X,\mu \right)\capL^{q}\left(X,\mu \right)$, where $\alpha =(r^{-1}-q^{-1})/(p^{-1}-q^{-1})$, $\beta =(p^{-1}-r^{-1})/(p^{-1}-q^{-1})$.

Let $\mu (X)<\infty$. Prove that $L^{\infty }\left(X,\mu \right)\subsetL^{p}\left(X,\mu \right)$ for all $p\geq 1$, and that $\left\| f\right\|_{\infty }=\lim_{p\to \infty }\left\| f\right\|_{p}$.

Prove that the following sets of functions are dense in $L^{p}\left({[a,b]}\right)$:

Find the norm of the function $f(x)=x^\alpha$ in those spaces $L^{p}\left({[0,1]}\right)$, $1\leq p\leq \infty$, to which it belongs.

In $L^{1}\left(\mathbb {R},dx\right)$ construct:

Let $\mu (X)<\infty$ and suppose that $V\subsetL^{1}\left(X,\mu \right)$ is a closed infinite-dimensional subspace. Prove that $V$ cannot be contained in $L^{\infty }\left(X,\mu \right)$.

Prove that the set $C_0(\mathbb {R})$ of continuous compactly supported functions is dense in $L^{p}\left(\mathbb {R},dx\right)$ for $1\leq p<\infty$.

Prove that every function $f\inL^{p}\left(\mathbb {R},dx\right)$ is continuous in the mean, i.e., for any $\varepsilon >0$ there is a $\delta >0$ such that for $\left|t\right|<\delta$

Prove the assertion of Problem 386 for the space $L^{p}\left(\mathbb {R}^{n},dx\right)$.

Let $M\subsetL^{p}\left(\mathbb {R}^{n},dx\right)$, $1\leq p<\infty$. Prove that $M$ is pre-compact if and only if:

1. There is a constant $c$ such that $\left\| f\right\|_{p}\leq c$ for all $f\in M$;

2. For any $\varepsilon >0$ there is a number $R(\varepsilon )$ such that

   $$\int _{\left|x\right|>R(\varepsilon )}\left|f(x)\right|\, dx<\varepsilon ,\qquad f\in M;$$

3. For any $\varepsilon >0$ there is a number $\delta (\varepsilon )>0$ such that for $\left|t\right|<\delta (\varepsilon )$

   $$\int _{\mathbb {R}^{n}}\left|f(x+t)-f(x)\right|^p\, dx<\varepsilon .$$

Verify the isomorphism of spaces

Find the extreme points of the unit ball in $L^{p}\left(X,\mu \right)$:

Prove that the spaces $L^{1}\left({[0,1]}\right)$ and $l_1$ are not isomorphic.

Prove that $C(X)$ is a Banach space for any compact space $X$.

Prove that $C(X)$ is a separable space if $X$ is a compact subset of $\mathbb {R}^{n}$.

Call a linear functional $F$ on $C(X)$ positive if $F(f)\geq 0$ for all nonnegative functions $f\in C(X)$. Prove that every positive linear functional $F$ is continuous, and its norm is equal to $F(1)$, where $1$ is the function identically equal to $1$ on $X$.

Prove that any functional $F\in C'(X)$ can be written in the form $F=F_1-F_2$, where $F_1$ and $F_2$ are positive functionals (see Problem 394).

Let $X$ be a compact metric space, and $F\in C'(X)$ a positive functional (see Problem 394). Define $\mu (K)=\inf_{\; \mathbb {1}_{K}\leq \varphi \leq 1}F(\varphi )$ for a compact set $K\subset X$, and $\mu (E)=\sup_{K\subset E}\mu (K)$ ($K$ compact) for a Borel set $E\subset X$. Prove that $\mu$ is a countably additive measure.

Prove that a sequence of functionals $F_n(f)=\int_0^1 f(x)\, dg_n(x)$, with $g_n\in BV[0,1]$, has a weak-* limit functional $F(f)=\int_0^1 f(x)\, dg(x)$ with $g\in BV[0,1]$ if and only if $g_n(x)\to g(x)$ at each point of $[0,1]$ and the variations of the functions $g_n$ are collectively bounded. Here $BV[0,1]$ is the space of functions of bounded variation on $[0,1]$.

Let $\mathcal P$ be the subspace of polynomials in $C[0,1]$. Which of the following linear functionals on $\mathcal P$ admit continuous extension to $C[0,1]$ ($p$ denotes the polynomial $\sum_{k=0}^{\deg p}a_k x^k$):

Let $X$ be a connected compact space. Prove that the unit ball in $C(X)$ has exactly two extreme points.

Prove that the signed point measures $\pm \mu_x$, $x\in X$, defined by the formula $\left\langle \mu_x, f \right\rangle =f(x)$ are the extreme points of the unit ball of $C'(X)$.

Let $X$ be a compact metric space, and $A\subset C(X)$ a closed subalgebra separating points (i.e., for any two distinct points $x_1$ and $x_2$ in $X$ there is a function $\varphi \in A$ such that $\varphi (x_1)\neq \varphi (x_2)$) and containing the function identically equal to $1$.

Let $X$ be an arcwise connected compact metric space. Construct a continuous mapping of $[0,1]$ onto $X$.

Construct a continuous mapping of $[0,1]$ onto the unit square.

Construct an isometric imbedding of $l_p(2,\mathbb {R})$ into $C[0,1]$ with the aid of a continuous mapping of $[0,1]$ onto the unit sphere of $l_p(2,\mathbb {R})$.

Prove that the spaces $C[0,1]\underset {\vee }{\otimes }C[0,1]$ and $C(\square )$ are isomorphic, where $\square$ is the unit square in $\mathbb {R}^{2}$.

Prove that there is an isomorphism $C(X)\underset {\vee }{\otimes }C(Y)\approx C(X\times Y)$ for any compact subsets $X$ and $Y$ of $\mathbb {R}^{n}$.

Let $A:C(X)\to C(Y)$ be an isomorphism of Banach spaces. Prove that $A$ has the form $(Af)(y)=a(y)f(\varphi (y))$, where $a$ is a continuous function on $Y$ taking the values $\pm 1$, and $\varphi$ is a homeomorphism of $Y$ onto $X$.

Prove that the space of all functions of the form $f(x)+g(y)$, where $f,g\in C[0,1]$, is closed in $C(\square )$ ($\square$ is the unit square in $\mathbb {R}^{2}$).

Prove that the space $C[0,1]$ has a countable topological basis $\left\{ f_n(x)\right\}$, i.e., a system of functions $\left\{ f_n(x)\right\}$ such that any $f\in C[0,1]$ can be uniquely represented as a uniformly convergent series $f(x)=\sum_{n=1}^{\infty }c_n f_n(x)$.

Prove that the system of functions $\left\{ e^{2\pi inx}\right\}$, $n\in \mathbb {Z}$, is not a topological basis (see Problem 410) in the space $CP[0,1]$ of all continuous functions on $[0,1]$ with the condition $f(0)=f(1)$.

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 $S(\mathbb {R}^{n})$.

Prove that the space $\mathcal D(\mathbb {R})$ of test functions is contained in the space $\mathcal D'(\mathbb {R})$ of generalized functions.

Does $\lim_{\varepsilon \downarrow 0}\sin (x/\varepsilon )$ exist in $\mathcal D'(\mathbb {R})$?

Let $f$ and $g$ be locally integrable functions on a domain $\Omega \subset \mathbb {R}^{n}$ defining the same regular generalized function (i.e., $\int_\Omega f\varphi \, dx=\int_\Omega g\varphi \, dx$ for all $\varphi \in \mathcal D(\Omega )$). Prove that $f$ and $g$ coincide almost everywhere.

Prove that the Dirac delta-function $\delta$, defined by $\int_{-\infty }^\infty \delta (x)\varphi (x)\, dx=\varphi (0)$, is not a regular generalized function.

Define generalized functions on $T^n$ as continuous linear functionals on $\mathcal D(T^n)$ (see Problem 423). Prove that the series $\sum_{k\in \mathbb {Z}^n}e^{2\pi ikt}$, where $t\in \mathbb {R}^{n}$, $kt=k_1t_1+\cdots +k_nt_n$, converges in $\mathcal D'(T^n)$ to the generalized function $\delta (t)$ defined by $\int_{T^n}\delta \varphi \, dt=\varphi (0)$.

Prove that every generalized function on $T^n$ (see Problem 435) has finite order, i.e., extends to a continuous linear functional on $C^k(T^n)$ of $k$-smooth functions on $T^n$ for some $k$.

Prove that the generalized function $F$ on the line defined by $\langle F,\varphi \rangle =\sum_{k=0}^\infty \varphi^{(k)}(k)$ does not have finite order.

Prove the Sokhotskii identity $1/(x\pm i0)=P\frac1x\mp \pi i\delta$.

Prove that the generalized functions $1/(x\pm i0)$ have order one on any bounded domain in $\mathbb {R}$ containing zero.

Let $\varphi \in \mathcal D(\mathbb {R})$.

Prove that every continuous linear mapping $A:\mathcal D(\Omega_1)\to \mathcal D'(\Omega_2)$ has the form

where $K\in \mathcal D'(\Omega_1\times \Omega_2)$.

In the conditions of Problem 442 let $\Omega_1=\Omega_2=\mathbb {R}$. Find explicitly the generalized function $K\in \mathcal D'(\mathbb {R}^{2})$ if the mapping $A$

Prove that a generalized function $F\in \mathcal D'(\mathbb {R})$ with the property that $F'=0$ is a constant.

Prove that all the solutions of the equation $xF=0$ among the generalized functions $F\in \mathcal D'(\mathbb {R})$ are proportional to the $\delta$-function.

Prove that every generalized function on the line with support at a point $a\in \mathbb {R}$ has the form $P(d/dx)\delta_a(x)$, where $P$ is a polynomial.

Suppose that $g\in \mathcal E(\mathbb {R})$ determines a one-to-one mapping of the line onto itself, and let $h\in \mathcal E(\mathbb {R})$ be the inverse mapping. Express the function $\delta '(g(x))$ explicitly in terms of $\delta (x)$ and its derivatives.

A generalized function $F$ on the line is said to be homogeneous of degree $(\lambda ,\varepsilon )$, where $\lambda \in \mathbb {C}$ and $\varepsilon =0$ or $1$, if $F(tx)=\left|t\right|^{\lambda }(\operatorname {sgn}t)^\varepsilon F(x)$ for $t\ne 0$.

Prove that a homogeneous function of degree $(\lambda ,\varepsilon )$ on the line (see first-edition Problem 513) satisfies the differential equation $xF'=\lambda F$.

Prove that for all $\lambda \in \mathbb {C}$ and $\varepsilon =0,1$ there exists a homogeneous generalized function on the line of degree $(\lambda ,\varepsilon )$, and that it is unique to within a numerical factor.

Prove that the only generalized functions on the line that are translation invariant are the constants.

Suppose that $F$ is a generalized function on the plane that is invariant under translations along the axis $Ox$.

Suppose that $F$ is a generalized function on the plane with support the segment $[0,1]$ of the axis $Ox$.