Problems to Solve

Let $G$ be a subspace of a Hilbert space $H$ and $f \in G^{*}$. Describe all continuous linear extensions $F$ of the functional $f$ to $H$. Which of them preserve the norm?

Solve problem (1) for $G = \left\{ x \in H \mid (x, h) = 0\right\}$ and $f(x) = (x, a), x \in G$, where $a, h \in H$ are fixed elements.

Solve problem (1) for $G = \left\{ x \in H \mid \left(x, h_{i}\right) = 0,1 \leq i \leq n\right\}$ and $f(x) = (x, a), x \in G$, where $a \in H$ is a fixed element, and $\left\{ h_{i}: 1 \leq i \leq n\right\} \subset H$ is a fixed orthonormal system.

$f\left(x_{1}\right) = 2, f\left(x_{2}\right) = 3$;

$f\left(x_{1}\right) = 2,\left\| f\right\| = 2, f\left(x_{2}\right) \leq 0$.

Let $G$ be dense in $X$. Prove that $G^{*} = X^{*}$.

Is the converse statement correct?

$f(x) = a_{0}$;

$f(x) = \sum_{k = 0}^{n} a_{k}$;

$f(x) = \sum_{k = 0}^{n}(-1)^{k} a_{k}$;

$f(x) = \sum_{k = 0}^{N} c_{k} a_{k}$, where $N \in \mathbb {N}$ and $c_{0}, \ldots , c_{N} \in \mathbb {R}$ are fixed numbers.

$F(x) = x(0)$ for every $x \in C(\left[-1,1\right])$;

$F(x) = x(0)$ for every $x \in L_{\infty }(\left[-1,1\right])$ such that $x$ is continuous at point 0 (i.e. there exists a continuous at point 0 function $\tilde{x}$ such that $x(t) = \tilde{x}(t)$ almost everywhere on $\left[-1,1\right]$);

$F(x) = 0$ for every $x \in L_{\infty }(\left[-1,1\right])$ such that $x$ is continuous at point 0 ;

$F(x) = 0$ for every $x \in L_{\infty }(\left[-1,1\right])$ such that $x = 0$ almost everywhere on $\left[-1,0\right]$ or the function $x$ is continuous at point 0.

Prove that there exists a functional $F \in l_{\infty }^{*}$ such that $F(x) = \lim_{n \rightarrow \infty } x_{n}$ for every $x \in c ;$

Prove that the inclusion $l_{1} \subset l_{\infty }^{*}$ is strict.

Let $l_{\infty }$ be a real Banach space. Prove that there exists a functional $f \in l_{\infty }^{*}$ such that

(a) $\left\| f\right\| = 1$;

(b) $f(e) = 1$ for $e = (1,1,1, \ldots )$;

(c) $f(T x) = f(x)$ for all $x \in l_{\infty }$, where $T\left(x_{1}, x_{2}, \ldots \right) = \left(x_{2}, x_{3}, \ldots \right)$. Any functional $f \in l_{\infty }^{*}$ with properties

(a)-(c) is called a Banach limit and is denoted by $\operatorname {Lim}_{n \rightarrow \infty } x_{n} = f(x), x \in l_{\infty }$.

Prove that $\operatorname {Lim}_{n \rightarrow \infty } x_{n}$ has the following properties:

1. if $x_{n} \geq 0, n \geq 1$, then $\operatorname {Lim}_{n \rightarrow \infty } x_{n} \geq 0$;

2. $\liminf_{n \rightarrow \infty } x_{n} \leq \operatorname {Lim}_{n \rightarrow \infty } x_{n} \leq \limsup_{n \rightarrow \infty } x_{n}$;

3. if $x_{n} \rightarrow a, n \rightarrow \infty$, then $\operatorname {Lim}_{n \rightarrow \infty } x_{n} = a$.

There exists $x \in l_{\infty }$ such that $x_{n} = 1$ for an infinite number of indices $n$, but $\operatorname {Lim}_{n \rightarrow \infty } x_{n} = 0$.

There exist $x, y \in l_{\infty }$ such that $\operatorname {Lim}_{n \rightarrow \infty } x_{n} y_{n} \neq \operatorname {Lim}_{n \rightarrow \infty } x_{n} \cdot \operatorname {Lim}_{n \rightarrow \infty } y_{n}$.

Let $B([0,+\infty ))$ be a linear space of bounded real functions on $[0,+\infty )$ with the norm $\left\| x\right\| = \sup_{t \geq 0}\left|x(t)\right|, \quad x \in B([0,+\infty ))$. Prove that there exists a functional $f \in (B([0,+\infty )))^{*}$ such that

(a) $\left\| f\right\| = 1$;

(b) $f(e) = 1$ for $e(t) = 1, t \geq 0$;

(c) $f\left(T_{s} x\right) = f(x)$ for all $x \in B([0,+\infty ))$ and $s > 0$, where $T_{s} x(t) = x(t+s), t \geq 0$.

For any functional $f \in (B([0,+\infty )))^{*}$ with properties (a)-(c) denote $\operatorname {Lim}_{t \rightarrow +\infty } x(t) = f(x), x \in B([0,+\infty ))$. Prove that $\operatorname {Lim}_{t \rightarrow +\infty } x(t)$ has the following properties:

1. if $x(t) \geq 0, t \geq 0$, then $\operatorname {Lim}_{t \rightarrow +\infty } x(t) \geq 0$;

2. $\liminf_{t \rightarrow +\infty } x(t) \leq \operatorname {Lim}_{t \rightarrow +\infty } x(t) \leq \limsup_{t \rightarrow +\infty } x(t)$;

3. if $x(t) \rightarrow a, t \rightarrow +\infty$, then $\operatorname {Lim}_{t \rightarrow +\infty } x(t) = a$.

Let $\left\{ x_{1}, \ldots , x_{n}\right\} \subset X$ be a linearly independent system. Prove that there exists a system biorthogonal to it.

Is the statement of item (1) correct for an infinite system of linearly independent elements of space $X$?

Let $\left\{ x_{\alpha }: \alpha \in A\right\} \subset X,\left\{ f_{\alpha }: \alpha \in A\right\} \subset X^{*}$ be biorthogonal systems. Is it correct that the system $\left\{ x_{\alpha }: \alpha \in A\right\} \subset X$ is linearly independent? Is it correct that the system $\left\{ f_{\alpha }: \alpha \in A\right\} \subset X^{*}$ is linearly independent?

Under what conditions on the linearly independent system $\left\{ x_{\alpha }: \alpha \in A\right\} \subset X$ does a biorthogonal system exist?

Let $\left\{ f_{1}, \ldots , f_{n}\right\} \subset X^{*}$ be a linearly independent system. Prove that there exists a system in $X$ that is biorthogonal to it.

for all $f \in X^{*}$?

for all $f$ from a total subset in $X^{*}$?

Let $X$ be an infinite-dimensional NLS. Show that $X^{*}$ is an infinite-dimensional space.

Is the converse statement to statement (1) true?

How are the dimensions of spaces $X$ and $X^{*}$ related, if both these spaces are finite-dimensional?

Every reflexive NLS is a Banach space?

If for a Banach space $X$ the set $\varphi (X)$ is dense in $X^{* *}$, where $\varphi : X \rightarrow X^{* *}$ is the canonical embedding from Theorem 4.3, then $X$ is reflexive?

with the help of the statement of problem 4.33.

with the help of the statement of problem 4.40*.

of Corollary 4.1 of the Hahn-Banach Theorem in $\mathbb {R}^{3}$, if $G$ is a line that passes through the origin;

of Corollary 4.2 of the Hahn-Banach Theorem in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$.

Prove that $M^{\perp }$ is a subspace in $X$.

What is $M^{\perp }$, if $X$ is a Hilbert space?

Prove that $\left\{ x \in X \mid f(x) = 0 \text{ for all } f \in M^{\perp }\right\} = \overline{\operatorname {span}(M)}$.

Prove that if $X$ is a reflexive space and $M \subset X$ is a subspace, then $M^{\perp \perp } = M$ (more precisely, $M^{\perp \perp } = \varphi (M)$, where $\varphi : X \rightarrow X^{* *}$ is the canonical embedding from Theorem 4.3).

Is the statement of item (4) valid for a nonreflexive space $X$?

Prove that $M^{\perp \perp } \cap X = \overline{\operatorname {span}(M)}$ (more precisely, $M^{\perp \perp } \cap \varphi (X) = \varphi (\overline{\operatorname {span}(M)})$).

Prove that (a) implies (b).

Prove that if $X$ is a reflexive Banach space then (b) implies (a).

Let $X$ be a dual space of some Banach space and (b) implies (a). Prove that the space $X$ is reflexive.

Prove that if $x \in C(\left[a, b\right])$ and $f(x) = 0$ for all $f \in M$, then $x = 0$, but the set $M$ is not total in $(C(\left[a, b\right]))^{*}$.

Find $\overline{\operatorname {span}(M)}$ in $(C(\left[a, b\right]))^{*}$.

Prove that in a separable NLS $X$ every closed ball is the intersection of a countable family of sets of the form $\left\{ x \in X \mid \left|f(x)-b\right| \leq a\right\}$, where $f \in X^{*}, a \in \mathbb {R}, b \in \mathbb {K}$.

Prove that in a real separable NLS $X$ every closed ball is the intersection of a countable family of sets of the form $\left\{ x \in X \mid f(x) \leq a\right\}$, where $f \in X^{*}, a \in \mathbb {R}$.

A subset $G \subset X$ is a hypersubspace if and only if there exists a functional $f \in X^{*} \backslash \left\{ 0\right\}$ such that $G = \operatorname {Ker} f$.

A subset $M \subset X$ is a hyperplane if and only if there exists a functional $f \in X^{*} \backslash \left\{ 0\right\}$ and $\lambda \in \mathbb {K}$ such that $M = \left\{ x \in X \mid f(x) = \lambda \right\}$.

If a functional $p: X \rightarrow \mathbb {R}$ is a seminorm, then $p(x) \geq 0, x \in X$;

If $p$ is a seminorm, then $\left|p(x)-p(y)\right| \leq p(x-y), x, y \in X$;

If $f$ is a linear functional on $X$, then the functional $p(x) = \left|f(x)\right|, x \in X$, is a seminorm;

If $p$ is a seminorm and $f$ is a linear functional on $X$ such that $f(x) \leq p(x), x \in X$, then $\left|f(x)\right| \leq p(x), x \in X$.

$p$ is not necessarily a seminorm;

the statement of problem 4.52, item (4) may be false for $p$.

If $f$ is a linear functional on $X$ for which $\left|f(x)\right| \leq p(x), x \in X$, and $g(x) = \mathrm{Re}\left(f(x)\right), h(x) = \mathrm{Im}\left(f(x)\right), x \in X$, then $g, h$ are linear functionals on $X_{\mathbb {R}}$ such that $\left|g(x)\right| \leq p(x),\left|h(x)\right| \leq p(x), x \in X$, and also $h(x) = -g(i x), x \in X$.

If $g$ is a linear functional on $X_{\mathbb {R}}$, for which $\left|g(x)\right| \leq p(x), x \in X$, and $f(x) = g(x)-i g(i x), x \in X$, then $f$ is a linear functional on $X$ such that $\left|f(x)\right| \leq p(x), x \in X$.

If $M_{0}$ is a real hyperplane, then $i M_{0}$ is also a real hyperplane in $X$.

The set $M \subset X$ is a hypersubspace in $X$ if and only if there exists a real hypersubspace $M_{0}$ in $X$ such that $M = M_{0} \cap \left(i M_{0}\right)$.

If the set $M \subset X$ is a hyperplane in $X$, then there exist real hyperplanes $M_{1}, M_{2}$ in $X$ such that $M = M_{1} \cap M_{2}$.

Let $M(\mathbb {R})$ be a linear space of bounded real functions on $\mathbb {R}$ with period 1, $p$ be a semi-additive positively homogeneous functional on $M(\mathbb {R})$, defined in problem 4.58, and $F: M(\mathbb {R}) \rightarrow \mathbb {R}$ be a linear functional such that $F(x) \leq p(x), x \in M(\mathbb {R})$ (such a functional exists according to problem 4.57). Prove that

(a) if $x(t) \geq 0, t \in \mathbb {R}$, then $F(x) \geq 0$;

(b) $F\left(T_{s} x\right) = F(x)$ for all $x \in M(\mathbb {R})$ and $s \in \mathbb {R}$, where

(c) $F(e) = 1$ for $e(t) = 1, t \in \mathbb {R}$;

(d) if $x(t) = \chi_{\left[a, b\right]}(t)$ for $t \in [0,1], 0 \leq a < b \leq 1$, then $F(x) = b-a$.

Prove that every linear functional $F$ on $M(\mathbb {R})$, which satisfies conditions (a)-(c), coincides with the Riemann integral $\int_{0}^{1} x(t) d t$ for Riemann integrable on $\left[0,1\right]$ functions $x$.

Prove that the functional $F$ which satisfies conditions (a)-(c) can be chosen in such a way that it coincides with the Lebesgue integral $\int_{0}^{1} x(t) d t$ for Lebesgue integrable on $\left[0,1\right]$ functions $x$.

Prove that it is possible to assign to every set $A \subset [0,1]$ a number $\mu (A)$, the "measure" of the set $A$, in such a way that for any sets $A \subset [0,1], B \subset [0,1]$ the following conditions hold true:

(a) if $A \cap B = \varnothing$, then $\mu (A \cup B) = \mu (A)+\mu (B)$;

(b) $\mu (A) \geq 0$;

(c) if a set $A$ can be transformed to a set $B$ by parallel translation, then $\mu (A) = \mu (B)$;

(d) $\mu (\left[0,1\right]) = 1$.

Let the set $A \subset [0,1]$ be Jordan measurable. Prove that $\mu (A)$ equals to the Jordan measure of the set $A$.

Prove that the functional $\mu : 2^{\left[0,1\right]} \rightarrow \mathbb {R}$, which satisfies conditions (a)-(d), can be chosen in such a way that it coincides with the Lebesgue measure for the Lebesgue measurable sets.