General Info
Remember: for every question you must indicate all the people with whom you collaborated. If you didn't collaborate with anyone else on a given problem, write "Collaborators: none."
Please write legibly, or type-up your solutions in LaTeX. The grader will greatly appreciate it.
This problem set will be due in class, Monday September 18.
Problems
(For the definition of a vector space, please refer to Wikipedia. Ditto definitions of rings and modules.)
- Verify that given a set \(\Omega\) and a vector space \(V\) over the field \(F\) of scalars, the set of functions \[ V^\Omega := \{ f: \Omega \to V\} \] form a vector space over the field \(F\) under point-wise operations.
- Verify that given a set \(\Omega\) and a field \(F\), the set of functions \[ F^\Omega := \{ f:\Omega \to F\} \] form a ring with point-wise operations. Identify the additive and multiplicative identities of \(F^\Omega\).
- Verify that \(V^\Omega\) is in fact a module over \(F^\Omega\).
- Verify that given \(\Omega\subset\mathbb{R}^d\) an open set, and \(W\) a normed real vector space, for any \(k\in \mathbb{N}\) the set \(C^k(\Omega; W)\) is a module over \(C^k(\Omega;\mathbb{R})\).
- Given that \(\Omega\) is an bounded open subset of \(\mathbb{R}^d\), verify that the expression \(|\cdot|_{C^k(\bar{\Omega}; W)}\) is a norm for \(C^k(\overline{\Omega};W)\) when regarded as a vector space over \(\mathbb{R}\).
- Consider the linear wave equation and the Laplace equation which we write as \[ \partial^2_{tt}u = \pm \partial^2_{xx} u \] with initial value prescribed on the set \( \{t = 0\}\) \[ u(0,x) = f(x), \qquad \partial_tu(0,x) = g(x).\] Suppose that \(f,g\in C^\omega\) have Maclaurin series with non-zero radii of convergence \[ f(x) = \sum_{k = 0}^\infty \frac{1}{k!} f_k x^k, \qquad g(x) = \sum_{k = 0}^\infty \frac{1}{k!} g_k x^k.\] Write down the Maclaurin series for \(u(t,x)\) about the point \((t,x) = (0,0)\) by solving for the coefficients in terms of \(f_k,g_k\), and verify that it has a non-zero radius of convergence.