Banach-Steinhaus Theorem. Weak convergence of vectors, functionals, and operators

(Part of the proof of Theorem 7.1). Let $X$ be a Banach space, $Y$ a normed space, and $\left\{ T_{n}\right\}_{n = 1}^{\infty } \subset \mathcal{B}(X, Y)$, with $\left\| T_{n}\right\| \rightarrow \infty$ as $n \rightarrow \infty$. Prove that for any $\lambda > 0$ the set

is nowhere dense in $X$.

Theorem 7.1 (S. Banach, H. Steinhaus, 1927): let $X$ be a Banach space, $Y$ a normed space, and $\mathcal{M} \subset \mathcal{B}(X, Y)$ a family of operators. If for every $x \in X$ $\sup_{A \in \mathcal{M}}\left\| A x\right\| \leqslant C(x)$, where the constant $C(x)$ depends only on the vector $x$, then there exists a constant $C_{1} > 0$ such that $\sup_{A \in \mathcal{M}}\left\| A\right\| \leqslant C_{1}$, i.e. the set $\mathcal{M}$ is bounded in the space $\mathcal{B}(X, Y)$.

Show that the completeness of the space $X$ in the statement of Theorem 7.1 is essential.

Theorem 7.1 (S. Banach, H. Steinhaus, 1927): let $X$ be a Banach space, $Y$ a normed space, and $\mathcal{M} \subset \mathcal{B}(X, Y)$ a family of operators. If for every $x \in X$ $\sup_{A \in \mathcal{M}}\left\| A x\right\| \leqslant C(x)$, where the constant $C(x)$ depends only on the vector $x$, then there exists a constant $C_{1} > 0$ such that $\sup_{A \in \mathcal{M}}\left\| A\right\| \leqslant C_{1}$, i.e. the set $\mathcal{M}$ is bounded in the space $\mathcal{B}(X, Y)$.

Let $f_{n} \in l_{\infty }$ be a sequence of functionals on the space $l_{1}, f_{n}(x) = n x_{n}, n = 1,2, \ldots$ This family of functionals is not bounded in the space $l_{\infty } = \left(l_{1}\right)^{*}$, and hence, by the Banach-Steinhaus theorem, there is a vector $x \in l_{1}$ such that $\sup \left|f_{n}(x)\right| = \infty$. Find such a vector $x$.

(E. Hellinger, O. Toeplitz, 1910). Let $A: H \rightarrow H$ be a linear operator in a Hilbert space $H$. Prove that if for all $x, y \in H$ the equality $(A x, y) = (x, A y)$ holds, then the operator $A$ is bounded. ¹

Definition 8.3: An operator $A \in \mathcal{B}(H)$ is called self-adjoint if $A = A^{*}$.

Let $X$ and $Y$ be Banach spaces. Prove that if a bilinear form $B(x, y)$, defined on $X \times Y$, is continuous separately in each argument, then it is continuous jointly in both arguments.

Define the operator

in the space $C[-\pi , \pi ]$. Prove that $\left\| S_{n}\right\| \rightarrow \infty$. Deduce from this that there exists a continuous function $x \in C[-\pi , \pi ]$ whose trigonometric Fourier series diverges at the point $t = 0$.

Prove that convergence in norm implies weak convergence. Give an example of a sequence in a Banach space that converges weakly but not in norm.

Prove that $x_{n} \rightarrow x$, where $\left\{ x_{n}\right\}$ and $x$ are vectors of a normed space $X$, if and only if $x_{n}$ converges weakly to $x$ uniformly on the unit ball of the space $X^{*}$, i.e.,

Let $x_{n} \rightharpoonup x, y_{n} \rightharpoonup y, \alpha_{n} \rightarrow \alpha , \beta_{n} \rightarrow \beta$, where $\left\{ x_{n}\right\} , x,\left\{ y_{n}\right\}$ and $y$ are vectors of a normed space $X$, and $\left\{ \alpha_{n}\right\} , \alpha ,\left\{ \beta_{n}\right\} , \beta$ are complex numbers. Prove that $\alpha_{n} x_{n}+\beta_{n} y_{n} \rightharpoonup \alpha x+\beta y$. Prove that if $x_{n} \rightharpoonup x$ and $x_{n} \rightharpoonup x^{\prime }$, then $x = x^{\prime }$.

(S. Mazur, 1933). Let a sequence $x_{n}$ converge weakly to $x$ in a normed space $X$. Prove that $x \in \overline{\operatorname {conv}\left\{ x_{n}\right\}_{1}^{\infty }}$.

(H. Hahn, 1922). Prove that in an arbitrary normed space $X$ any weakly bounded set is bounded in norm. Prove that weak convergence or weak fundamentality of a sequence of vectors of a normed space implies boundedness of this sequence in norm.

Prove that on bounded sets of a separable Hilbert space weak convergence is equivalent to coordinatewise convergence, namely $x_{n} \rightharpoonup x$ if and only if $\left(x_{n}, e_{k}\right) \rightarrow \left(x, e_{k}\right)$ for every $k \in \mathbb {N}$, where $\left\{ e_{k}\right\}_{k = 1}^{\infty }$ is an arbitrary fixed orthonormal basis.

Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X, Y)$. Prove that if $x_{n} \rightharpoonup x$ in the space $X$, then $A x_{n} \rightharpoonup A x$ in the space $Y$. Give an example showing that the sequence $\left\{ A x_{n}\right\}_{1}^{\infty }$ need not converge to $A x$ in norm.

Prove the criterion for weak convergence. Let $X$ be a normed space, and $\left\{ x_{n}\right\}_{n = 1}^{\infty }$ a sequence of vectors in it. Prove that $x_{n} \rightharpoonup x$ if and only if two conditions hold:

1. the sequence is bounded, i.e., there is a number $C$ such that $\left\| x_{n}\right\| < C$ for all $n \in \mathbb {N}$;

2. in $X^{*}$ there is an everywhere dense set $Y$ such that $f\left(x_{n}\right) \rightarrow f(x)$ for every $f \in Y$.

Prove that in a finite-dimensional normed space weak convergence coincides with convergence in norm, i.e.,

Prove the criterion for weak convergence in $l_{p}, p \in (1, \infty )$: a sequence $x^{n} = \left(x_{1}^{n}, x_{2}^{n}, \ldots \right)$ converges weakly to $x = \left(x_{1}, x_{2}, \ldots \right)$ if and only if two conditions hold:

1. there is a number $C$ such that $\left\| x^{n}\right\|_{l_{p}} < C$ for all $n \in \mathbb {N}$;

2. $\lim_{n \rightarrow \infty } x_{k}^{n} = x_{k}$ for all $k \in \mathbb {N}$.

(J. Schur, 1909). Prove that weak convergence in $l_{1}$ coincides with convergence in norm (uniform convergence).

Prove that the space $l_{1}$ has no subspaces isomorphic to the space $l_{2}$.

Prove the criterion for weak convergence in $c_{0}$: a sequence $x^{n} = \left(x_{1}^{n}, x_{2}^{n}, \ldots \right)$ converges weakly to $x = \left(x_{1}, x_{2}, \ldots \right)$ if and only if two conditions hold:

1. there is a number $C$ such that $\left\| x^{n}\right\|_{c_{0}} < C$ for all $n \in \mathbb {N}$;

2. $\lim_{n \rightarrow \infty } x_{k}^{n} = x_{k}$ for all $k \in \mathbb {N}$.

Prove the criterion for weak convergence in $c$: a sequence $x^{n} = \left(x_{1}^{n}, x_{2}^{n}, \ldots \right)$ converges weakly to $x = \left(x_{1}, x_{2}, \ldots \right)$ if and only if three conditions hold:

1. there is a number $C$ such that $\left\| x^{n}\right\|_{c} < C$ for all $n \in \mathbb {N}$;

2. $\lim_{n \rightarrow \infty } x_{k}^{n} = x_{k}$ for all $k \in \mathbb {N}$;

3. if we denote $l_{n} = \lim_{k \rightarrow \infty } x_{k}^{n}$ and $l = \lim_{k \rightarrow \infty } x_{k}$, then $l_{n} \rightarrow l$.

Prove the criterion for weak convergence in $C[0,1]$: a sequence $x_{n} = x_{n}(t)$ converges weakly to $x$ if and only if two conditions hold:

1. there is a number $C$ such that $\left\| x\right\|_{C[0,1]} < C$ for all $n \in \mathbb {N}$;

2. $\lim_{n \rightarrow \infty } x_{n}(t) = x(t)$ for every $t \in [0,1]$.

Prove the criterion for weak convergence in $L_{p}[0,1], p \in [1, \infty )$: a sequence $x_{n} = x_{n}(t)$ converges weakly to $x$ if and only if two conditions hold:

1. there is a number $C$ such that $\left\| x_{n}\right\|_{L_{p}} < C$ for all $n \in \mathbb {N}$;

2. for every element of an arbitrary system of functions $\left\{ g_{\alpha }\right\} \subset L_{q}[0,1] = \left(L_{p}[0,1]\right)^{*}$ satisfying the condition $\overline{\operatorname {Lin}\left\{ g_{\alpha }\right\} } = L_{q}[0,1]$, the equality holds

   $$\lim _{n \rightarrow \infty } \int _{0}^{1} x_{n}(t) g_{\alpha }(t) d t = \int _{0}^{1} x(t) g_{\alpha }(t) d t$$

   In Chapter 10 it will be proved that for $p > 1$ as the system $\left\{ g_{\alpha }\right\}$ one can take any of the systems

   $$\left\{ t^{m}\right\} _{m = 0}^{\infty }, \quad \left\{ \sin \pi m t\right\} _{m = 1}^{\infty }, \quad \left\{ \chi _{(a, b)}(t): a \in [0,1], b \in [0,1], a < b\right\}$$

   (here $\chi_{(a, b)}$ is the characteristic function of the interval $(a, b)$). In the case $p = 1$ and $q = \infty$, as the system $\left\{ g_{\alpha }\right\}$ one can choose the system of characteristic functions of all Lebesgue measurable sets on $[0,1]$.

Prove that in a Hilbert space $x_{n} \rightarrow x$ if and only if $x_{n} \rightharpoonup x$ and $\left\| x_{n}\right\| \rightarrow \left\| x\right\|$.

Prove that in any uniformly convex normed space (see Definition 2.11) $x_{n} \rightarrow x$ if and only if $x_{n} \rightharpoonup x$ and $\left\| x_{n}\right\| \rightarrow \left\| x\right\|$.

Definition 2.11: A normed space $X$ is called uniformly convex if for every $\varepsilon > 0$ there exists $\delta > 0$ such that if $\left\| x\right\| = \left\| y\right\| = 1$ and $\left\| x+y\right\| > 2-\delta$, then $\left\| x-y\right\| < \varepsilon$.

Prove that weak convergence in the space $L_{p}[0,1]$, $p \in [1, \infty )$, does not in general imply convergence in measure.

Prove that if $p > 1$ and a sequence of functions $x_{n} \in L_{p}[0,1]$, bounded in the norm of $L_{p}[0,1]$, converges in measure to some function $x$, then $x \in L_{p}[0,1]$ and $x_{n} \rightharpoonup x$. Prove that for $p = 1$ this statement is false.

Prove that if $x_{n} \rightharpoonup 0$ in a normed space $X$, then $\underline{\lim }_{n \rightarrow \infty }\left\| x-x_{n}\right\| \geqslant \left\| x\right\|$ for every element $x$.

In the spaces $c_{0}$ and $C[0,1]$ give examples of weakly fundamental sequences having no weak limit. $\square$

Prove that every reflexive Banach space is sequentially weakly complete (i.e., every weakly fundamental sequence has a weak limit). Give an example of a non-reflexive but sequentially weakly complete Banach space.

Let $H$ be a Hilbert space, $\left\{ x_{n}\right\}_{1}^{\infty }$ and $\left\{ y_{n}\right\}_{1}^{\infty }$ sequences of vectors from $H$. Will the numerical sequence $\left(x_{n}, y_{n}\right)$ be convergent if the sequences $\left\{ x_{n}\right\}_{1}^{\infty }$ and $\left\{ y_{n}\right\}_{1}^{\infty }$ are fundamental (in norm or weakly)? Investigate all three cases.

Let $X$ be a normed space, and $X^{*}$ separable (hence, by the assertion of problem 5.42, the space $X$ itself is separable). Prove that weak convergence on the unit ball of the space $X$ is metrizable.

Let $X$ be a normed space, and $X^{*}$ separable. Specify a metric $w(x, y)$, defined on all of $X$, such that a sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ converges weakly to $x \in X$ if and only if two conditions hold:

1. the sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ is bounded;

2. $w\left(x_{n}, x\right) \rightarrow 0$.

(S. Banach, 1932). Let a Banach space $X$ be reflexive, and $X^{*}$ separable. Prove that from any sequence of points of the unit ball $\left\{ x:\left\| x\right\|_{X} \leqslant 1\right\}$ one can select a subsequence that converges weakly to a point of the unit ball (that is, the unit ball is sequentially weakly compact).

Let a Banach space $X$ be reflexive, and $X^{*}$ separable. Prove that any bounded set $B \subset X$ is weakly sequentially precompact, i.e., from any sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ of points of the set $B$ one can extract a weakly convergent subsequence $x_{n_{k}} \rightarrow x \in X$. Prove that if, moreover, $B$ is a closed and convex set, then $x \in B$, i.e., $B$ is weakly sequentially compact.

Prove that the unit balls in the spaces $l_{1}, c_{0}, C[0,1]$ are not weakly sequentially compact sets.

Prove that any sequence of nested nonempty closed convex bounded sets in a reflexive Banach space has nonempty intersection. Give a counterexample to this statement for the case of non-reflexive Banach spaces (compare with problems 1.36, 1.38, 2.68, 3.34, and 4.22).

Let $f_{n} \xrightarrow {*} f, g_{n} \xrightarrow {*} g, \alpha_{n} \rightarrow \alpha , \beta_{n} \rightarrow \beta$, where $\left\{ f_{n}\right\} , f,\left\{ g_{n}\right\}$ and $g$ are functionals from the dual space $X^{*}$, and $\left\{ \alpha_{n}\right\} , \alpha ,\left\{ \beta_{n}\right\}$, $\beta$ are complex numbers. Prove that $\alpha_{n} f_{n}+\beta_{n} g_{n} \xrightarrow {*} \alpha f+\beta g$. Prove that if $f_{n} \xrightarrow {*} f$ and $f_{n} \xrightarrow {*} \tilde{f}$, then $f = \tilde{f}$.

Prove that convergence in norm of the dual space implies weak-* convergence. Give an example of a sequence in the dual space that converges weak-*, but does not converge in norm.

Prove that bounded linear functionals $f_{n} \in X^{*}$ converge in norm to a functional $f$ if and only if they converge weak-* to $f$ uniformly on the unit ball of the space $X$ (i.e., for every $\varepsilon > 0$ there is $N$ such that the inequality $\left|\left(f_{n}-f\right)(x)\right| < \varepsilon$ holds for all $x \in X,\left\| x\right\| \leqslant 1$, and $n > N$).

Prove that weak convergence in the dual space implies weak-* convergence. Give an example of a sequence in the dual space that converges weak-*, but does not converge weakly.

Let $X$ be a normed space with $\operatorname {dim} X < \infty$. Prove that then all three types of convergence (in norm, weak, and weak-*) in the space $X^{*}$ coincide.

Let $X$ be a reflexive normed space. Prove that weak and weak-* convergence in the space $X^{*}$ coincide.

(H. Hahn, 1922). Prove that any weak-* bounded set in $X^{*}$ is bounded in norm. Prove that weak-* convergence or weak-* fundamentality of a sequence of functionals from $X^{*}$ implies boundedness of this sequence in norm.

Let $X$ be a normed space. Prove that the space $X^{*}$ is complete with respect to weak-* convergence (i.e., every weak-* fundamental sequence of functionals has a weak-* limit in $X^{*}$).

Give an example of a sequence $f_{n} \xrightarrow {*} f$ such that $f \notin \overline{\operatorname {Lin}\left\{ f_{n}\right\}_{1}^{\infty }}$.

Let $X$ be a separable normed space. Prove that weak-* convergence on the unit ball of the space $X^{*}$ is metrizable.

Let $X$ be a separable normed space. Specify a metric $w(x, y)$, defined on all of $X^{*}$, such that a sequence $\left\{ f_{n}\right\}_{1}^{\infty } *$-weak converges to $f \in X^{*}$ if and only if two conditions hold:

1. the sequence $\left\{ f_{n}\right\}_{1}^{\infty }$ is bounded;

2. $w\left(f_{n}, f\right) \rightarrow 0$.

Let $X$ be a separable normed space. Prove that the unit ball in $X^{*}$ is weak-* sequentially compact, i.e., from any sequence $\left\{ f_{n}\right\}_{1}^{\infty },\left\| f_{n}\right\|_{X^{*}} \leqslant 1$, one can select a subsequence that converges weak-* to a point of the unit ball.

Let $X$ be a normed space, and $\left\{ f_{n}\right\}_{1}^{\infty }$ a sequence of functionals from $X^{*}$. Prove that $f_{n} \xrightarrow {*} f$ if and only if two conditions hold:

1. the sequence is bounded, i.e., there is a number $C$ such that $\left\| f_{n}\right\|_{X^{*}} < C$ for all $n \in \mathbb {N}$;

2. in $X$ there is an everywhere dense set $Y$ such that $f_{n}(x) \rightarrow f(x)$ for every $x \in Y$.

Prove the criterion for weak-* convergence in $l_{1} = \left(c_{0}\right)^{*}$: a sequence $x_{n} = \left(x_{1}^{(n)}, x_{2}^{(n)}, \ldots \right)$ converges weak-* to $x = \left(x_{1}, x_{2}, \ldots \right)$ if and only if two conditions hold:

1. there is a number $C$ such that $\left\| x_{n}\right\|_{l_{1}} < C$ for all $n \in \mathbb {N}$;

2. $\lim_{n \rightarrow \infty } x_{k}^{(n)} = x_{k}$ for all $k \in \mathbb {N}$.

Prove the criterion for weak-* convergence in $l_{\infty } = \left(l_{1}\right)^{*}$: a sequence $x_{n} = \left(x_{1}^{(n)}, x_{2}^{(n)}, \ldots \right)$ converges weak-* to $x = \left(x_{1}, x_{2}, \ldots \right)$ if and only if two conditions hold:

1. there is a number $C$ such that $\left\| x_{n}\right\|_{l_{\infty }} < C$ for all $n \in \mathbb {N}$;

2. $\lim_{n \rightarrow \infty } x_{k}^{(n)} = x_{k}$ for all $k \in \mathbb {N}$.

Prove the criterion for weak-* convergence in the space $B V_{0}[0,1] = (C[0,1])^{*}$: a sequence $g_{n}$ converges weak-* to $g$ if and only if two conditions hold:

1. there is a number $C$ such that $\left\| g_{n}\right\|_{B V_{0}[0,1]} < C$ for all $n \in \mathbb {N}$;

2. for every element of an arbitrary system of functions $\left\{ f_{\alpha }\right\}$ in the space $C[0,1]$ satisfying the condition $\overline{\operatorname {Lin}\left\{ f_{\alpha }\right\} } = C[0,1]$, the equality holds

   $$\lim _{n \rightarrow \infty } \int _{0}^{1} f_{\alpha }(t) d g_{n} = \int _{0}^{1} f_{\alpha }(t) d g$$

Let $A = \left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$ be an infinite lower triangular matrix. Let $\sum_{j = 1}^{\infty } x_{j}$ be an arbitrary series. Consider the method of summation of this series defined by the matrix $A$ according to the rule $S(A) = \lim_{n \rightarrow \infty }\left(\sum_{j = 1}^{n} a_{n j} x_{j}\right)$. Prove that a necessary condition for the correctness ¹ of this summation method is the condition

For a numerical sequence $\left\{ x_{k}\right\}_{1}^{\infty }$ define a generalized limit by means of an arbitrary infinite matrix $A = \left\{ a_{i, j}\right\}_{i, j = 1}^{\infty }$, infinite to the right and downward:

Prove that $\sigma (A, x) = \lim x_{k}$ for every sequence $\left\{ x_{k}\right\}_{1}^{\infty } \in c$ if and only if three conditions hold:

1. $\lim_{n \rightarrow \infty } a_{n k} = 0, k = 1,2, \ldots$;

2. $\lim_{n \rightarrow \infty } \sum_{k = 1}^{\infty } a_{n k} = 1$;

3. there is a number $M$ such that $\sum_{k = 1}^{\infty }\left|a_{n k}\right| \leqslant M$ for all $n$.

(N. Abel, 1826). Prove the correctness of Abel's summation method. Let the series $\sum_{k = 1}^{\infty } a_{k}$ converge to a number $a$. Prove that $\lim_{x < 1, x \rightarrow 1} \sum_{k = 1}^{\infty } a_{k} x^{k} = a$.

Let $X$ be a normed space, $\left\{ x_{n}\right\}_{1}^{\infty }$ a sequence of vectors from $X$, and $\left\{ f_{n}\right\}_{1}^{\infty }$ a sequence of functionals from $X^{*}$. Will the numerical sequence $f_{n}\left(x_{n}\right)$ be convergent if the sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ is fundamental (in norm or weakly) and the sequence $\left\{ f_{n}\right\}_{1}^{\infty }$ is fundamental (in norm, weakly, or weak-*)? Investigate all six cases.

Determine the type of convergence (in norm, weak, weak-* if it concerns the dual space — or none) of the following sequences, and in case of convergence find the limit:

Let $X$ be a Banach space. Prove that for any operator $A \in \mathcal{B}\left(X, c_{0}\right)$ there is a sequence of functionals $\left\{ f_{n}\right\}_{1}^{\infty } \subset X^{*}$ such that $f_{n} \rightharpoonup 0$ and $(A x)_{n} = f_{n}(x)$ for every $x \in X$. Prove that, conversely, any sequence of functionals $\left\{ f_{n}\right\}_{1}^{\infty } \subset X^{*}$ converging weakly to zero generates an operator $A \in \mathcal{B}\left(X, c_{0}\right)$ by the rule $(A x)_{n} = f_{n}(x)$.

Let $X$ and $Y$ be normed spaces, sequences of operators $A_{n}$ and $B_{n}$ converging strongly in $\mathcal{B}(X, Y)$ to $A$ and $B$ respectively, and sequences of complex numbers $\alpha_{n}$ and $\beta_{n}$ converging to $\alpha$ and $\beta$ respectively. Prove that $\alpha_{n} A_{n}+\beta_{n} B_{n} \xrightarrow {s} \alpha A+\beta B$. Prove that if $A_{n} \xrightarrow {s} A$ and $A_{n} \xrightarrow {s} B$ in $\mathcal{B}(X, Y)$, then $A = B$. Prove both statements of the problem with strong operator convergence replaced by weak operator convergence.

Prove that uniform convergence of a sequence of operators from $\mathcal{B}(X, Y)$ implies strong operator convergence, and strong operator convergence implies weak operator convergence.

Give an example of a sequence of operators of the space $\mathcal{B}(X, Y)$ that converges strongly, but not uniformly. Give an example of a sequence of operators that converges weakly, but not strongly.

Prove that a sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty }$ from $\mathcal{B}(X, Y)$ converges in norm to an operator $A$ if and only if $A_{n} \xrightarrow {s} A$ uniformly on the unit ball of the space $X$, i.e., for every $\varepsilon > 0$ there is a number $N$ such that for every $x \in X$, $\left\| x\right\|_{X} \leqslant 1$ and for all $n > N$ the inequality $\left\| A_{n} x-A x\right\|_{Y} \leqslant \varepsilon$ holds. Prove that a sequence $\left\{ A_{n}\right\}_{1}^{\infty }$ converges in norm to an operator $A$ if and only if $A_{n} \rightharpoonup A$ uniformly on the unit ball of the space $X$ and the unit ball of the space $Y^{*}$, i.e., for every $\varepsilon > 0$ there is a number $N$ such that for all $x \in X$, $\left\| x\right\|_{X} \leqslant 1$ and $f \in Y^{*},\left\| f\right\|_{Y^{*}} \leqslant 1$ and for all $n > N$ the inequality $\left|f\left(A_{n} x-A x\right)\right| \leqslant \varepsilon$ holds.

Let $X$ and $Y$ be normed spaces, with $X$ finite-dimensional. Prove that in the space $\mathcal{B}(X, Y)$ uniform and strong operator convergence coincide.

Let $X$ and $Y$ be normed spaces, with $Y$ finite-dimensional. Prove that in the space $\mathcal{B}(X, Y)$ strong and weak operator convergence coincide.

Let $X$ be a normed space, and $Y$ a Banach space. Prove that the space $\mathcal{B}(X, Y)$ is complete with respect to strong operator convergence.

Let $X$ and $Y$ be normed spaces and let $Y$ be complete with respect to weak convergence. Prove that the space $\mathcal{B}(X, Y)$ is complete with respect to weak operator convergence.

(H. Hahn, 1922). Let $X$ be a Banach space, and $Y$ a normed space. Let $\mathcal{M} \subset \mathcal{B}(X, Y)$ be a weakly bounded set of operators, i.e., for all $x \in X$ and $f \in Y^{*}$ the inequalities $\sup_{A \in \mathcal{M}}\left|f(A x)\right| < C_{f, x} < \infty$ hold. Prove that $\mathcal{M}$ is bounded in norm. ¹ In particular, any weakly convergent and any weakly fundamental sequence is bounded in norm.

Let a sequence of continuous operators $A_{n}$ in a Banach space $X$ converge weakly to $A \in \mathcal{B}(X)$. Prove that $\left\| A\right\| \leqslant \varlimsup_{n \rightarrow \infty }\left\| A_{n}\right\|$. Give an example showing that even in the case of strong operator convergence equality may fail.

Let $X, Y$ and $Z$ be normed spaces, let a sequence $\left\{ A_{n}\right\}_{1}^{\infty } \subset \mathcal{B}(X, Y)$ converge (uniformly, strongly, or weakly) to an operator $A \in \mathcal{B}(X, Y)$, and let a sequence $\left\{ B_{n}\right\}_{1}^{\infty } \subset \mathcal{B}(Y, Z)$ converge (uniformly, strongly, or weakly) to an operator $B \in \mathcal{B}(Y, Z)$. In each of the nine possible cases determine whether the sequence $B_{n} A_{n}$ converges to the operator $B A$, and indicate the type of convergence.

Prove the criterion for strong convergence of operators. Let $X$ and $Y$ be Banach spaces. A sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty }$ in $\mathcal{B}(X, Y)$ converges strongly to an operator $A$ if and only if two conditions hold:

1. for every $x \in X$ the sequence $\left\{ A_{n} x\right\}_{1}^{\infty }$ is bounded;

2. there exists an everywhere dense set $M$ in $X$ such that $A_{n} x \rightarrow A x$ for every $x \in M$.

Prove the criterion for weak convergence of operators. Let $X$ and $Y$ be Banach spaces. A sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty }$ in $\mathcal{B}(X, Y)$ converges weakly to an operator $A$ if and only if two conditions hold:

1. for all $x \in X$ and $f \in Y^{*}$ the sequence $\left\{ f\left(A_{n} x\right)\right\}_{1}^{\infty }$ is bounded;

2. there exists an everywhere dense set $M$ in $X$, and an everywhere dense set $L$ in $Y^{*}$, such that $f\left(A_{n} x\right) \rightarrow f(A x)$ for all $x \in M$ and $f \in L$.

Let $X$ and $Y$ be Banach spaces, let the spaces $X$ and $Y^{*}$ (and hence also the space $Y$) be separable, and let $Y$ be reflexive. Prove that weak operator convergence on any bounded set of the space $\mathcal{B}(X, Y)$ is metrizable (compare with problems 7.31 and 7.46).

Let $X$ and $Y$ be Banach spaces, let the spaces $X$ and $Y^{*}$ (and hence also the space $Y$) be separable, and let $Y$ be reflexive. Prove that from any bounded sequence of operators from $\mathcal{B}(X, Y)$ one can extract a weakly convergent subsequence (i.e., any bounded set in $\mathcal{B}(X, Y)$ is sequentially precompact with respect to weak operator convergence; compare with problems 7.34 and 7.48).

Give an example showing that the statement of the previous problem is false for the case of strong operator convergence.

For the following sequences of operators $A_{n}$, determine the type of convergence (uniform, strong, weak, or none) and find the limit in the indicated spaces: