# HW1 (MTH847 Fall 2017)

### 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."

1. 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.
2. 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 $$F^\Omega$$.
3. Verify that $$V^\Omega$$ is in fact a module over $$F^\Omega$$.
4. 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 $$C^k(\Omega;\mathbb{R})$$.
5. 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 $$\mathbb{R}$$.
6. 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.