Set of 2-Dimensional Indefinite Real Orthogonal Matrices is not Compact in Normed Real Square Matrix Vector Space

Theorem
Let $\struct {\R^{2 \times2}, \norm {\, \cdot \,}_\infty}$ be the normed real matrix vector space.

Let $\map O {1, 1} := \set {\mathbf R \in \R^{2 \times2} : \mathbf R^\intercal \mathbf J_{1,1} \mathbf R = \mathbf J_{1,1}}$ be the indefinite orthogonal group of degree $\paren {1, 1}$ over real numbers where:


 * $\ds \mathbf J_{1,1} := \begin{bmatrix}

1 & 0\\ 0 & -1 \\ \end{bmatrix}$

Then $\map O {1, 1}$ is not a compact set in $\struct {\R^{2 \times 2}, \norm {\, \cdot \,}_\infty}$.

Proof
Let:


 * $\begin{bmatrix}

\map \cosh t& \map \sinh t\\ \map \sinh t & \map \cosh t \\ \end{bmatrix} := \map {\mathbf R} t$

We have that:

Hence, $\map {\mathbf R} t \in \map O {1, 1}$.

From difference of squares of hyperbolic cosine and sine:

The set of matrix elements constitutes a finite subset of the set of real numbers which is ordered.

By Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements, there is a maximal element.

Consider the supremum norm of $\map {\mathbf R} t$:

Furthermore:


 * $\ds \lim_{t \mathop \to \infty} \map \cosh t = \infty$

Hence, $\map O {1,1}$ is not bounded.

By Heine-Borel theorem, $\map O {1,1}$ is not compact in $\struct {\R^{2 \times 2}, \norm {\, \cdot \,}_\infty}$.