Weak Convergence

Let $X$ be a normed linear space and suppose that $x_{n} \rightharpoonup x_{0} \in X$. Prove that $\left\| x_{0}\right\| \leq \liminf \left\| x_{n}\right\|$.

A sequence $\left\{ x_{n}\right\}$ in a normed linear space $X$ is weakly Cauchy if $\Lambda \left(x_{n}\right)$ is a Cauchy sequence for every $\Lambda \in X^{*}$. The space $X$ is weakly complete if every weakly Cauchy sequence in $X$ is weakly convergent. Prove that every reflexive Banach space is weakly complete.

Prove that a subspace of a normed linear space is closed if and only if it is weakly closed.

Let $X, Y$ be Banach spaces and let $T: X \rightarrow Y$ be a linear operator. Prove that $T$ is bounded if and only if whenever $\left\{ x_{n}\right\} \subset X$ converges weakly to $x$, then $\left\{ T x_{n}\right\}$ converges weakly to $T x$.

Prove that a weakly convergent sequence can have only one limit.

Let $X, Y$ be Banach spaces and let $\left\{ T_{n}\right\}$ be a bounded sequence in $\mathcal{L}(X, Y)$. Let $\mathcal{M}$ be a set of all $x \in X$ for which every subsequence of $\left\{ T_{n} x\right\}$ has a convergent subsequence. Prove that $\mathcal{M}$ is a closed subspace of $X$.

Let $X$ be a normed linear space. Prove that every weakly compact subset of $X$ is norm-bounded.

Let $X$ be a normed linear space. Prove that the weak topology is generated by the sets of the form $U_{a, r, T_{1}, \ldots , T_{n}}:=\left\{ x \in X:\left|T_{k}(x)-T_{k}(a)\right|<r, 1 \leq k \leq n, T_{k} \in X^{*}\right\}$.

Let $X$ be an infinite dimensional normed linear space. Prove that every weakly open subset of $X$ is unbounded.

Let $X$ be a normed linear space and $T$ a linear functional on $X$. Prove that $T$ is continuous in the weak topology if and only if it is continuous in the norm topology.

Let $\mathcal{H}$ be Hilbert space and $T, T_{1}, T_{2}, \ldots$ bounded linear operators on $\mathcal{H}$ such that $\lim_{n} \left\langle T_{n} x, y \right\rangle = \left\langle T x, y \right\rangle$ for all $x, y \in \mathcal{H}$. Prove that there exists $C>0$ s.t. $\left\| T_{n}\right\| \leq C$ for all $n \in \mathbb {N}$.

Prove that in a finite-dimensional normed linear space, every weakly convergent sequence is strongly convergent.

Prove Theorem 12.6.5.

Theorem 12.6.5: A closed and bounded set in $L^{p}\left([0,1]\right)$ is weakly sequentially compact, if $1 < p < \infty$.

Prove the assertions made in Example 12.6.7.

Example 12.6.7 (a sequence in $L^{p}$ that converges in measure but not weakly): Let $f_{n} = n\chi_{[0,1/n]}$, for $n \in \mathbb {N}$. Then $\left\{ f_{n}\right\}$ converges in measure on $[0,1]$ but it does not converge weakly in $L^{p}\left([0,1]\right)$.

Prove the assertions made in Example 12.6.8.

Example 12.6.8 (a sequence in $L^{p}$ that converges uniformly but not weakly): Let $f_{n} = \frac{1}{n}\chi_{[1,e^{n}]}$, for $n \in \mathbb {N}$. Then $\left\{ f_{n}\right\}$ converges uniformly on $\mathbb {R}$, but it does not converge weakly in $L^{p}\left(\mathbb {R}\right)$ for any $p \in [1, +\infty )$.

Give an example of a measure space and a sequence of functions in $L^{p}\left(X, \mu \right)$ that converges weakly but not in measure.

Prove Theorem 12.6.11.

Theorem 12.6.11: Let $(X, \mathfrak {M}, \mu )$ be a measure space. A sequence $\left\{ f_{n}\right\}$ in $L^{p}\left(X, \mu \right)$, $1 \leq p < \infty$, converges weakly to $f_{0}$ if and only if both: (a) There exists $M > 0$ such that $\left\| f_{n}\right\|_{p} \leq M$, for all $n \in \mathbb {N}$. (b) $\lim_{n} \int_{E} f_{n} \, d\mu = \int_{E} f_{0} \, d\mu$, for every $E \in \mathfrak {M}$ that has finite measure.

The purpose of this exercise is to establish that conditions (a)-(b) of Theorem 12.6.13 implies the strong convergence of $\left\{ f_{n}\right\}$ to $f_{0}$. We will follow the original proof of Riesz.

Prove or disprove: If a sequence $\left\{ f_{n}\right\}$ converges weakly in $L^{p}\left({[0,1]}\right)$ then it has a subsequence that converges a.e. in $[0,1]$.

Let $\left\{ f_{n}\right\}$ be a sequence in $L^{2}[0,2 \pi ]$ defined by $f_{n}(x)=\sin n x$. Prove that $\left\{ f_{n}\right\}$ converges weakly but not strongly.

Let $\left\{ f_{n}\right\}$ be a sequence in $L^{2}[0,1]$ that converges in measure and suppose there is $M>0$ such that $\left\| f_{n}\right\| \leq M$, for all $n \in \mathbb {N}$. Prove that $\left\{ f_{n}\right\}$ is weakly convergent. Does the result hold for $L^{p}[0,1]$ for $1<p<\infty$ ?

Let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ be a sequence of nonnegative functions in $L^{2}\left(0,1\right)$, and suppose that $\left\{ f_{n}\right\}$ converges to a function $f$ in the norm of $L^{2}\left(0,1\right)$. Prove that $f \geq 0$. Does the statement remain true if $\left\{ f_{n}\right\}$ converges weakly to $f$ ?

Give an example of a measure space ( $X, \mathfrak {M}, \mu$ ) and a bounded sequence $\left\{ f_{n}\right\}$ in $L^{1}\left(X, \mu \right)$ such that $\left\{ f_{n}\right\}$ converges to $f$ a.e. but does not converge weakly to $f$ in $L^{1}\left(X, \mu \right)$.

Let $\left\{ f_{n}\right\}$ be a sequence in $L^{p}[0,1], 1<p<\infty$. Prove that $\left\{ f_{n}\right\}$ converges weakly to $f_{0}$ if and only if both:

Prove that the closed unit ball of $L^{1}[0,1]$ is not weakly sequentially compact.

Give an example of a measure space ( $X, \mathfrak {M}, \mu$ ) for which the unit ball of $L^{1}\left(X, \mu \right)$ is weakly compact. Is it possible to take $X \subset \mathbb {R}, \mathfrak {M}=\mathcal{M} \mid X$, and $\mu =m \mid X$ ?

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p<\infty$, let $1 / p+1 / q=1$, and suppose that $\left\{ f_{n}\right\}$ converges weakly to $f$ in $L^{p}\left(X, \mu \right)$ and $\left\{ g_{n}\right\}$ converges to $g$ in the norm of $L^{q}\left(X, \mu \right)$. Prove that

Let $h \in C(\mathbb {R})$ be a periodic function with period $T$, let $\int_{0}^{T} h=0$, let $1 \leq p<\infty$, and let $f_{n}:[a, b] \rightarrow \mathbb {R}$ be defined by $f_{n}(x)=h(n x)$ for $n \in \mathbb {N}$. Prove that $\left\{ f_{n}\right\}$ converges weakly to 0 in $L^{p}\left({[a, b]}\right)$.

Let $1<p<\infty$, let $h \in L^{p}\left(\mathbb {R}\right)$, and let $f_{n}: \mathbb {R} \rightarrow \mathbb {R}$ be defined by $f_{n}(x)=h(x-n)$ for $n \in \mathbb {N}$. Prove that $\left\{ f_{n}\right\}$ converges weakly to 0 in $L^{p}\left(\mathbb {R}\right)$. Is this true for $p=1$ ?

Let $1 \leq p<r<\infty$, let $\left\{ f_{n}\right\} , f \in L^{r}[0,1]$. Prove or disprove:

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1<p<\infty$, let $\left\{ f_{n}\right\} , f \in L^{p}\left(X, \mu \right)$, and suppose that the sequence $\left\{ f_{n}\right\}$ is bounded in $L^{p}\left(X, \mu \right)$. Consider the following four properties:

1. $\lim_{n} f_{n}=f$ a.e.

2. $\left\{ f_{n}\right\}$ converges weakly to $f$ in $L^{p}\left(X, \mu \right)$.

3. $\lim_{n}\left\| f_{n}\right\|_{p}=\left\| f\right\|_{p}$.

4. $\left\{ f_{n}\right\}$ converges to $f$ in the norm of $L^{p}\left(X, \mu \right)$.

If $\left\{ f_{n}\right\}$ possesses two of these properties, is there a subsequence that possesses all four properties?

Give an example to show that Theorem 12.6.13 is not true in $L^{1}[0,1]$.

Theorem 12.6.13 (the Radon-Riesz property): Let $(X, \mathfrak {M}, \mu )$ be a measure space. A sequence $\left\{ f_{n}\right\}$ in $L^{p}\left(X, \mu \right)$, $p>1$, converges strongly (i.e., in norm) to $f_{0}$ if and only if both: (a) $f_{n} \rightharpoonup f_{0}$ (weak convergence). (b) $\lim_{n} \left\| f_{n}\right\| = \left\| f_{0}\right\|$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1<p<\infty$, let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ be a bounded sequence in $L^{p}\left(X, \mu \right)$, and suppose that $\left\{ f_{n}\right\}_{n=1}^{\infty }$ converges to $f$ a.e. Prove that $\left\{ f_{n}\right\}_{n=1}^{\infty }$ converges weakly to $f$ in $L^{p}\left(X, \mu \right)$.

Prove that every orthonormal sequence in Hilbert space converges weakly to 0 .

Let $X$ be an inner product space. Prove that $X$ is complete if and only if every bounded sequence has a weakly convergent subsequence.

Let $\alpha \in \mathbb {R}, \alpha \neq 1$, and let $\left\{ x_{n}\right\}$ be a sequence in Hilbert space such that

Prove that $\left\{ x_{n}\right\}$ is weakly convergent.

Let $\left\{ x_{n}\right\}$ be an orthogonal sequence in Hilbert space. Prove that the following are equivalent:

Find the weak closure of the unit sphere in Hilbert space.

Let $\left\{ e_{n}\right\}_{n=1}^{\infty }$ be an orthonormal basis of Hilbert space. Prove that $x_{n}$ converges weakly to $x$ if and only if $\left\{ x_{n}\right\}$ is bounded and for each $k$, $\left\{ \left\langle x_{n}, e_{k} \right\rangle \right\}$ converges to $\left\{ \left\langle x, e_{k} \right\rangle \right\}$. Give an example to show the boundedness of $\left\{ x_{n}\right\}$ cannot be omitted.

Prove Theorem 12.6.14.

Theorem 12.6.14: Let $\left\{ x_{n}\right\}$ converge weakly to $x$ in Hilbert space $H$. Then $\left\{ x_{n}\right\}$ converges to $x$ in the norm if and only if $\lim \left\| x_{n}\right\| = \left\| x\right\|$.

Give an example of a (strongly) closed set that is not weakly closed

Let $\left\{ f_{n}\right\}$ be a sequence in $C(X)$. Prove that $\left\{ f_{n}\right\}$ converges weakly to some function $f$ if and only if $\left\{ f_{n}\right\}$ is norm bounded and it converges pointwise on $X$.

Prove that the closed unit ball of $\ell^{p}, 1<p<\infty$, is weakly sequentially compact.

The purpose of this exercise is to establish that every weakly convergent sequence in $\ell^{1}$ is strongly convergent.

Let $\left\{ e_{n}\right\}$ be a sequence in $c_{0}$, such that for every $n \in \mathbb {N}, e_{n}$ has 1 in the $n$th spot and 0 elsewhere: $e_{n}=(0,0, \ldots , 0,1,0,0, \ldots )$. Prove that the sequence $\left\{ e_{n}\right\}$ converges weakly to 0 in $c_{0}$ and in $\ell^{p}$, for $1<p<\infty$, but not in $\ell_{1}$.

Prove that the closed unit ball of $\ell^{1}$ is not weakly sequentially compact.

Prove that there is no Banach space whose dual is isomorphic to $c_{0}$.

Let $\left\{ a_{k}\right\}$ be a sequence of nonzero numbers and