Open Mapping and Closed Graph Theorems

State the open mapping theorem for Banach spaces and use it to prove the following: Let \((B, \|\cdot \|_{B})\) and \((C, \|\cdot \|_{C})\) be Banach spaces and \(A : B \to C\) be continuous and linear. If \(A\) is a bijection then \(A^{-1}\) is continuous.

1. State the Closed Graph Theorem.
2. Let \(T : U \to V\) be a linear map between two Banach spaces such that, for any bounded linear functional \(f\) on \(V\) , the composite \(f \circ T\) is a bounded functional on \(U\) . Prove that \(T\) is bounded. You may use the fact that bounded linear functionals on a Banach space separate points.

Let \(T:U\to V\) be a linear map between two Banach spaces such that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use the closed graph theorem.

Let \(T:U\to V\) be a linear map between two Banach spaces such that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use closed graph theorem.

Let \(f:\Omega \to \mathbb C\) be a complex-valued function on an open domain \(\Omega \subset \mathbb C\). Assume that for all closed contours \(\gamma \) that consists of line segments parallel to the real or imaginary axes \[ \int _\gamma f(z)\,dz=0. \] Show that there exists a holomorphic function \(F:\Omega \to \mathbb C\) such that \(F'(z)=f(z)\).

The space \(C[0,1]\) of continuous functions on \([0,1]\) is a subspace of all the \(L^p\) spaces, \(L^p[0,1]\) for \(1\le p\le \infty \). Define a linear operator \(T:C[0,1]\to \mathbb R\) by \(Tf=f(1/2)\). Show that \(T\) is a bounded operator for the \(L^p\)-norm if and only if \(p=\infty \). Show that \(T\) extends to a bounded operator on all of \(L^\infty [0,1]\).

In what follows, \(D_R(z_0)\) and \(\overline {D_R}(z_0)\) denote the open and closed discs, respectively, centered at \(z_0\), with radius \(R\).

Consider \[ HC=\{(x_n)\in \mathbb C^{\mathbb Z}\mid |x_n|\le \frac {1}{n}\text { for all }n\in \mathbb Z\}. \] Note that \(HC\) is simply a product of (infinitely many) closed discs. As such, it is compact with respect to the product topology. The product topology on \(HC\) is defined so that sets \[ U_x(m,\delta )=\{(y_n)\in HC\mid |x_n-y_n|<\delta \text { for all }1\le n\le m\} \] provide a fundamental system of neighborhoods of \(x\). Next, note that \(HC\) can be considered a subset of the Hilbert space \(\ell ^2(\mathbb Z)\). Let \(B(x,\epsilon )\) be, as usual, the open ball of radius \(\epsilon \) in \(\ell ^2(\mathbb Z)\). Show that for every \(\epsilon >0\) there exists a positive integer \(m\) and \(\delta >0\) such that \[ U_x(m,\delta )\subseteq B(x,\epsilon )\cap HC \] and, for every positive integer \(m\) and \(\delta >0\) there exists \(\epsilon >0\) such that \[ U_x(m,\delta )\supseteq B(x,\epsilon )\cap HC \] In other words, show that the restricted topology on \(HC\) coincides with the product topology.

Let \(f\) be a holomorphic function defined on a neighbourhood of the closed disc \(|z|\le 1\). Assume that \(f(0)=1\), and \(|f(z)|>1\) if \(|z|=1\). Must \(f\) have a zero in the open disc \(|z|<1\)? Justify your answer.

a. Let \(\Omega \) be an open set containing the closed unit disk, \(f\) holomorphic on \(\Omega \), and suppose that \(f\) has an analytic logarithm on \(\Omega \). Show that if \(z=re^{i\theta }\) with \(0\le r<1\) then \[ \ln |f(z)|=\frac {1}{2\pi }\int _0^{2\pi }P_r(\theta -t)\ln |f(e^{it})|\,dt; \] \(P_r(\theta )\) denoting the Poisson kernel.

b. Let \(g\) be holomorphic on \(\Omega \) and assume that \(g(z)\ne 0\) for all \(|z|=1\). If \(a_1,\ldots ,a_n\) are the zeros of \(g\) in the open unit disk, repeated according to their multiplicities, \(z=re^{i\theta }\), \(0\le r<1\), \(g(z)\ne 0\), then show that \[ \ln |g(z)|=\sum _{j=1}^{n}\ln \left |\frac {z-a_j}{1-\bar a_j z}\right | +\frac {1}{2\pi }\int _0^{2\pi }P_r(\theta -t)\ln |g(e^{it})|\,dt. \]

a. Let \(\Re \alpha <0\), and \(g(z)=\frac {\alpha -z}{\bar \alpha +z}\). Show that \(g\) is a conformal map of \(\{z\mid z\in \mathbb C\text { and }\Re z<0\}\) onto the open unit disk. (Hint: the ordered triple \(\{\alpha ,\frac {\alpha -\bar \alpha }{2},-\bar \alpha \}\) maps to \(\{0,1,\infty \}\) under \(g\).)

b. Let \(f\) be holomorphic on the closed unit disk and suppose that \(\Re f(z)<0\) for all \(z\) in the closed disk. Then show that \[ |f'(0)|\le 2|\Re f(0)|. \]