6710-2023-08

Let \(X\) be a real Banach space and \(X^*\) denote its dual. Let \(C \subset X\) be a convex set containing the origin. Define

The goal of this problem is to prove that \(C^{**} = \overline {C}\).

1. Show that \(C^{**}\) is closed.
2. Show that \(\overline {C} \subset C^{**}\).
3. Show that \(\overline {C}\) is convex.

4. Assume there is \(x_0 \in X\) such that \(x_0 \in C^{**}\) but \(x_0 \notin \overline {C}\). Show there exists an \(f \in X^*\) such that for all \(x \in \overline {C}\) we have:

   \[f(x) < 1 < f(x_0).\]
5. Explain the contradiction in (d) and how it can be used to conclude with the desired result.

Let \(X\) be a Banach space over the complex numbers. Let \(X_{\mathbb {R}}\) be the vector space over the reals obtained by restricting the multiplication by scalars used to define \(X\) to the real line. Let \(f : X \to \mathbb {C}\) be a bounded linear functional on \(X\) and let \(\phi : X \to \mathbb {R}\) be its real part, defined by \(\phi (x) = \operatorname {Re} f(x)\) for all \(x \in X\).

1. Prove that \(\phi \) is a linear functional on \(X_{\mathbb {R}}\).

2. Prove that \(f\) can be reconstructed from \(\phi \) by:

   \[f(x) = \phi (x) - i\phi (ix), \quad \text {for } x \in X.\]

3. Show that \(f\) as a linear functional on \(X\) and \(\phi \) as a linear functional on \(X_{\mathbb {R}}\) have the same norm, i.e.

   \[\|f\|_{X^*} = \|\phi \|_{X_{\mathbb {R}}^*},\]

   where \(X^*\) is the dual of \(X\) and \(X_{\mathbb {R}}^*\) is the dual of \(X_{\mathbb {R}}\). Hint: To prove one of the inequalities, it may be helpful to find \(\lambda \in \mathbb {C}\) such that \(|f(x)| = f(x/\lambda )\) for a fixed \(x\) with \(f(x) \neq 0\).

Let \((\lambda _j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell ^2 \to \ell ^2\) defined by its action on a sequence \((\xi _j) \in \ell ^2\) by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence of finite rank operators \((T_k)\) that converges to \(T\) in an appropriate sense.

Let \(X\) be a Banach space. Let \(y \in X\) be fixed and let \(f : X \to \mathbb {R}\) be a bounded linear functional. Show that there is a constant \(C > 0\) such that for all scalars \(\alpha \) with \(|\alpha | < C\) the equation

has a unique solution on the open ball \(\{x \in X : \|x - y\| < 1\}\).