Weak Topologies and Weak Convergence

Is \(\{v\in \ell ^2(\mathbb {N}): |v_i|\le 1/\log (i+4)\}\) compact in the norm topology on \(\ell ^2(\mathbb {N})\)? Justify your answer.

Define the weak topology on \(L^{2}(\mathbb {R}, \lambda )\) and prove that it is not metrizable.

1. Give the definition of what it means for a sequence \((x_{n})\) in a Banach space \(X\) to converge weakly to \(x \in X\).
2. Let \(X = [0, 1]\) equipped with the Lebesgue measure. For \(n =\) 1, 2, …, let \(f_{n} = n \cdot \chi _{[0,1/n]}\), where \(\chi _{A}\) denotes the indicator function for the set \(A\). Prove that the sequence \(f_{n}\) does not converge weakly to 0 in \(L^{1}([0, 1])\).

Let \(f_n\in C([0,1])\) for \(n=1,2,\ldots \). Show that the following statements are equivalent:

(a) For every \(\lambda \in C([0,1])^*\) we have \(\lambda (f_n)\to 0\) as \(n\to \infty \).

(b) \(f_n(x)\to 0\) for every \(x\in [0,1]\) and \(\sup \lVert f_n\rVert _\infty <\infty \).

Here \(\lVert f\rVert _\infty =\sup _{x\in [0,1]}|f(x)|\).

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Let \(X=[0,1]\) with the usual measure. For \(n=1,2,\ldots \), let \(f_n=n^{1/p}\cdot \chi _{[0,1/n]}\). Prove that the sequence \(f_n\) does not converge weakly to \(0\) in \(L^1([0,1])\).

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Let \(X=[0,1]\) with the usual measure. Let \(p\ge 1\). For \(n=1,2,\ldots \), let \[ f_n=n^{1/p}\cdot \chi _{[0,1/n]}. \] Prove that the sequence \(f_n\) converges weakly to \(0\) in \(L^p([0,1])\) if and only if \(p>1\). Use that \[ L^p([0,1])^*\cong L^q([0,1]). \]

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Notation: \(D=\{z\in \mathbb C:|z|<1\}\).

Show that if \(v_i\in \mathcal H\) and \(v_i\) converges weakly to \(u\), then \[ \liminf _{i\to \infty }\lVert v_i\rVert \ge \lVert u\rVert , \] and that equality occurs only when there exists a subsequence \(v_{n_j}\) so that \(v_{n_j}\) converges to \(u\) in norm.

Prove that for every Hilbert space, the set of isometries is closed in the strong operator topology. Show that there exist Hilbert spaces where the set of isometries is not closed in the weak operator topology.

Consider \(\ell ^4(\mathbb N)\) equipped with topology induced by the norm \(\lVert \cdot \rVert _4\). Show that there exists an open set in this topology that is not open in the weak topology on \(\ell ^4(\mathbb N)\).

If \(v_i\in \mathcal H\) converges weakly to \(v_\infty \) and \(\lVert v_\infty \rVert =\lim _{i\to \infty }\lVert v_i\rVert \), then does \(v_i\) converge in norm to \(v_\infty \)?

Consider \(L^2(\mathbb Z):=H\). Recall two topologies on the set of bounded linear operators on \(H\): The norm topology which is metric topology given by \[ d(A,B)=\sup _{v\in H:\lVert v\rVert =1}\lVert Av-Bv\rVert . \] The strong operator topology which is given by \(A_i\to A_\infty \) if for every \(v\in H\) we have \[ \lim _{i\to \infty }\lVert A_iv-A_\infty v\rVert =0. \]

(a) Show that if \(A_i\) converges to \(A_\infty \) in the norm topology then it converges to \(A_\infty \) in the strong operator topology.

(b) Give an example of \(A_i\) which converges to \(A_\infty \) in the strong operator topology but not the norm topology.

(c) Let \(A_i\) be the operator defined by \(A_iv=w\) where \(w_j=v_j\) if \(j\ne i\) and \(w_i=iv_i\). Show \(A_1,\ldots \) is a sequence of operators that does not converge in either topology.

Find the Taylor series of the function \[ f(z)=\frac {z}{2+z} \] at the point \(a=1\). What is the radius of convergence of that series?

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous functions on \(X\), equipped with the sup norm: \(\|f\|_\infty =\sup _{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda \) on \(C(X)\), \(\lambda (f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup _n\|f_n\|_\infty \) is finite.

Let \(A\) be a closed subspace of \(C([0,1])\) which is also closed in \(L^2([0,1])\). Prove that the orthogonal projection \(P\) which projects \(L^2([0,1])\) onto \(A\) is continuous as a linear transformation from \(L^2([0,1])\) into \(C([0,1])\). Note that \(C([0,1])\) has the topology determined by the norm \(\|\cdot \|_\infty \), while \(L^2([0,1])\) has the topology determined by the norm \(\|\cdot \|_2\).

Suppose that \(f\) is holomorphic in the open disc of radius \(r\), centered at \(z_0\), but does not have an analytic extension to any neighborhood of the closure of this disc. Show that the radius of convergence of the power series expansion for \(f\) centered at \(z_0\) is \(r\).

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous functions on \(X\), equipped with the sup norm: \(\|f\|_\infty =\sup _{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda \) on \(C(X)\), \(\lambda (f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup _n\|f_n\|_\infty \) is finite.