Linear Topological Spaces

Prove that the topology in an LTS is determined by specifying the system of neighborhoods of zero.

Prove that in any LTS a closed set $X$ and a point $x\notin X$ have disjoint neighborhoods.

Suppose that two norms $p_1(x)$ and $p_2(x)$ are given in a finite-dimensional linear space. Prove that there is a positive constant $C$ such that

Prove that there is only one topology in a finite-dimensional linear space $L$ with respect to which $L$ is a Hausdorff LTS.

Prove that if two norms $p(x)$ and $q(x)$ majorize one another ($C^{-1}p(x)\leq q(x)\leq Cp(x)$, $C>0$), then the systems of open balls $\mathring B_p$ and $\mathring B_q$ determine the same topology.

Give an example of closed sets $A$ and $B$ for which $A+B$ is not closed.

Let $A_1,\ldots ,A_n$ be convex sets and $\lambda_1,\ldots ,\lambda_n$ fixed numbers. Prove that $A=\sum_{i=1}^n\lambda_i A_i$ is a convex set.

Prove that the intersection of any family of convex sets is convex.

Let $A$ be any bounded set in the plane and $B$ the unit disk $x^2+y^2<1$. Prove that for any positive numbers $\alpha$ and $\beta$ the set $\alpha A+\beta B$ is Lebesgue-measurable, and if $A$ is convex, then

What is the meaning of the coefficients $S$ and $L$?

Let $A_1,\ldots ,A_k$ be convex bounded sets in $\mathbb {R}^{n}$. Prove that $\mu \left(\alpha_1 A_1+\cdots +\alpha_k A_k\right)$ is a homogeneous polynomial of degree $n$ in the variables $\alpha_1,\ldots ,\alpha_k$.

Let $L$ be a linear space over the field $K$. Prove that among all Hausdorff topologies turning $L$ into an LCS there is a strongest, called the convex core topology, and that it has the following properties:

Prove that a set $B$ is the unit ball for some seminorm $p$ if and only if it is convex, balanced, and closed in the convex core topology (see Problem 278).

Let $A$ be any set in a linear space $L$, $c_1(A)$ the intersection of all the convex sets in $L$ containing $A$, and $c_2(A)$ the collection of all vectors of the form $x=\sum_{i=1}^n\tau_i x_i$, where $x_i\in A$ and the coefficients $\tau_i$ are nonnegative and have the property that $\sum_{i=1}^n\tau_i=1$. Prove that $c_1(A)=c_2(A)$. This set is called the convex hull of $A$ and denoted by $c(A)$.

A set $M$ in an LTS $L$ is said to be bounded if for each neighborhood $U$ of zero there is an $\varepsilon >0$ such that $\varepsilon M\subset U$. Prove that in a polynormed space $(L,\left\{ p_\alpha \right\}_{\alpha \in A})$ a set $M$ is bounded if and only if it is bounded in each of the seminorms $p_\alpha$, $\alpha \in A$.

Prove that in every LCS the convex hull of a bounded set is bounded. Give an example showing that this property may fail to hold in an arbitrary LTS.

Let $\mathbb {R}^{\infty }$ be the space of all sequences of real numbers with the topology of coordinatewise convergence (cf. Problem 288). Prove that there are no nonempty open bounded sets in $\mathbb {R}^{\infty }$.

In the space $C(\mathbb {R})$ of all continuous real functions on the line introduce the countable system of seminorms $p_N(f)=\max_{\left|x\right|\leq N}\left|f(x)\right|$ and set

For what $r>0$ is the ball of radius $r$ a convex set in $C(\mathbb {R})$?

In the space $C(\mathbb {R})$ of all continuous real functions on the line define a distance by the formula

Is $C(\mathbb {R})$ an LTS with the topology determined by this distance?

Prove that the subspace $BC(\mathbb {R})$ of the space $C(\mathbb {R})$ in the preceding problem consisting of all the bounded continuous functions on the line is a normed LTS.

Let $\mathbb {R}^{\infty }$ denote the space of all sequences of real numbers with the distance

Prove that:

Prove that if a linear functional is continuous at some point of a linear topological space, then it is continuous at each point of this space.

Let $L$ be a linear topological space. Prove that:

Suppose that the linear topological space $L$ has a defining system of neighborhoods of zero whose cardinality does not exceed the dimension of $L$. Prove that there is a discontinuous linear functional on $L$.

A convex set $M$ in an LTS $L$ for which the set $\left\{ x\in M\mid \forall y\in L\ \exists \varepsilon (y)\in \mathbb {R}:x+ty\in M\text{ for }\left|t\right|<\varepsilon (y)\right\}$ is nonempty is called a convex body. Convex bodies $B$ and $B'$ in $\mathbb {R}^{n}$ are said to be dual if their Minkowski functionals determine structures of dual normed spaces in $\mathbb {R}^{n}$. (We identify the vector $(a_1,\ldots ,a_n)$ with the functional $(x_1,\ldots ,x_n)\mapsto \sum_{i=1}^n a_i x_i$.) Prove that the section of $B$ by a $k$-dimensional plane $P$ is dual to the projection of $B'$ on this plane.

Prove that there does not exist a normed space whose dual space is $C[a,b]$.

Prove that if an LTS $L_1$ is Hausdorff and finite-dimensional, then every linear mapping $L_1\to L_2$, where $L_2$ is an arbitrary LTS, is continuous.

Let $P$ be the space of all polynomials in $x$ with real coefficients, and $U_+$ (resp., $U_-$) the subset of polynomials with positive (resp., negative) leading coefficient. Prove that the sets $U_+$ and $U_-$ are convex, but cannot be separated by a hyperplane.

Prove that disjoint convex closed sets $A$ and $B$ can be strictly separated by a hyperplane if one is compact. (This means that there exist a continuous linear functional $f$ and constants $c_1<c_2$ such that $f(x)\leq c_1$ on $A$ and $f(x)\geq c_2$ on $B$.)

Let $L$ be a normed linear space with norm $p$, $L'$ the dual space with norm $p'$, and $L''$ the space dual to $L'$, with norm $p''$. Assign to each $x\in L$ an element $F_x\in L''$ according to the formula $F_x(f)=f(x)$ for $f\in L'$. Prove that $p''(F_x)=p(x)$.

Let $L$ be a finite-dimensional normed space. Prove that $L$ and $L''$ are isomorphic (i.e., there exists a linear isometric mapping of $L$ onto $L''$).

Prove that every normed linear space is isometric to a subspace of some space of the form $C(X)$, where $C(X)$ is the space of continuous functions on a compact set $X$, with the norm $\left\| f\right\| =\max_{x\in X}\left|f(x)\right|$.

Construct an isometric imbedding of $l_p(2,\mathbb {R})$ into $C[0,1]$ for $p=1,2,\infty$.

Prove that there exists an isometric imbedding of $l_p(n,\mathbb {R})$ into $l_\infty (\mathbb {R})$.

Let $l_\infty (\mathbb {R})$ be the space of bounded real sequences $\left\{ x_n\right\}$, $n=1,2,\ldots$. Prove that there exists a linear functional $\operatorname {LIM}\in l_\infty (\mathbb {R})'$ having the following properties:

1. $\sup x_n\geq \operatorname {LIM}\left\{ x_n\right\} \geq \inf x_n$;

2. If $\lim_{n\to \infty }x_n=a$ exists, then $\operatorname {LIM}\left\{ x_n\right\} =a$;

3. $\operatorname {LIM}\left\{ x_{n+1}\right\} =\operatorname {LIM}\left\{ x_n\right\}$.

Prove that the assertions of Problem 303 carry over to the case of two-sided sequences $\left\{ x_n\right\}$, $n\in \mathbb {Z}$.

Let $L$ be a normed space, and $T$ a linear invertible operator on $L$ having the property that

Prove that there is a norm $\widetilde p$ in $L$ that is equivalent to $p$, and with respect to which $T$ is an isometry.

Let $L$ be an LCS. Prove that it can be imbedded continuously in a product of lines $\mathbb {R}^{\alpha }$, where $\alpha$ is a sufficiently large cardinal number. (In other words, every LCS admits a coordinate description.)

Let $B(\mathbb {R}^{n})$ be the space of bounded real functions on $\mathbb {R}^{n}$, with the norm $\left\| f\right\| =\sup_{x\in \mathbb {R}^{n}}\left|f(x)\right|$. Prove that there exists a linear functional $\operatorname {LIM}\in B(\mathbb {R}^{n})'$ having the properties:

1. $\inf_{\mathbb {R}^{n}}f(x)\leq \operatorname {LIM}f(x)\leq \sup_{\mathbb {R}^{n}}f(x)$;

2. If $\lim_{\left|x\right|\to \infty }f(x)=a$ exists, then $\operatorname {LIM}f(x)=a$;

3. $\operatorname {LIM}f(x+y)=\operatorname {LIM}f(x)$ for any $y\in \mathbb {R}^{n}$.

Prove that every closed convex set in a real LCS $L$ is the intersection of some family of half-spaces of the form $f(x)\leq c$, where $f\in L'$, $c\in \mathbb {R}$.

Represent the unit ball of $L_p(n,\mathbb {R})$ as an intersection of countably many half-spaces.

Let $I^N$ be the $N$-dimensional cube defined in $\mathbb {R}^{N}$ by the inequalities $\left|x_i\right|\leq 1$, $1\leq i\leq N$. Prove that any convex bounded subset of the plane can be approximated to any degree of accuracy by two-dimensional sections of $I^N$. (More precisely, for any $\varepsilon >0$ and any convex set $V\subset \mathbb {R}^{2}$ there exist an $N$ and an imbedding $\varphi :\mathbb {R}^{2}\to \mathbb {R}^{N}$ such that the Minkowski functionals of the sets $U$ and $V=\varphi^{-1}(I^N)$ are connected by the inequalities $1-\varepsilon <p_U/p_V<1+\varepsilon$.)

Construct an isometric imbedding of $l_p(n,\mathbb {R})$ into $C[0,1]$ for any $p\in [1,\infty ]$.

Prove that if a family of convex subsets of $\mathbb {R}^{n}$ is such that any $n+1$ sets of the family have a common point, then all the sets of the family have a common point.

Define a topology in $C[0,1]$ by taking the sets $U_\varepsilon =\left\{ f\in C[0,1]:\int_0^1\sqrt{\left|f(x)\right|}\, dx<\varepsilon \right\}$ as a base of neighborhoods of zero. Prove that each continuous linear functional on the resulting topological space is equal to zero.

Let $L$ be an LCS, $X$ a set with a measure $\mu$, and $f$ a function on $X$ with values in $L$. The function $f$ is said to be weakly measurable if the numerical function $F(f(x))$ is $\mu$-measurable for every $F\in L'$. An element $\varphi \in L$ is called a weak integral of $f$ with respect to the measure $\mu$ on $X$ if $F(f)=\int_K F(f(x))\, d\mu (x)$ for all $F\in L'$. Prove: