Which of the following subsets are dense in the given spaces?

Question

Which of the following subsets are dense in the given spaces?

• (a) The set of trigonometric polynomials in the space of continuous functions on $[−\pi, \pi]$ which are $2\pi$-periodic (with the sup-norm topology).

• (b) The subset of $C^ ∞$ functions with compact support in $\mathbb{R}$ in the space of bounded real-valued continuous functions on $\mathbb{R}$ (with the sup-norm topology).

• (c) $GL(n;\mathbb{R})$ in $M(n;\mathbb{R})$ (with its usual topology after identification with $\mathbb{R}^{n^2}$ ).

$GL(n;\mathbb{R})$ maps $\mathbb{R}^*$ by determinant function which is dense, so (c) is true. But how can I able to verify (a) and (b)?

Answer 1

For (a): it is true, just apply Fejér's theorem.

For (b): it is false. Let $f \in C_b^0(\mathbb{R})$ be identically equal to $1$ and $g \in C_c^{\infty}(\mathbb{R})$. Then, there exists $x \in \mathbb{R}$ such that $f(x)=0$, hence $||f-g||_{\infty} \geq |f(x)-g(x)|=1$.

For (c): it is true. If $A \in M_n(\mathbb{R})$, use Gauss elimination to show that there exist $P,Q \in GL_n(\mathbb{R})$ such that $A=P \left( \begin{array}{cc} I_r & 0 \\ 0 & 0 \end{array} \right)Q^{-1}$ where $r= \text{rank}(A)$. Then, if $A_m= P \left( \begin{array}{cc} I_r & 0 \\ 0 & \frac{1}{m}I_{n-r} \end{array} \right)Q^{-1}$, notice that $A_m \in GL_n(\mathbb{R})$ and $A_m \underset{m \to + \infty}{\longrightarrow} A$.