This was written in Page 92, of Higson's Analytic $K$-Homology book.
Let $H$ be a hilbert space. The $C^*$ algebra $K(H)$ of compact operators is the direct limit of a sequence $$M_2(\Bbb C) \subseteq M_4(\Bbb C) \subseteq \cdots $$
How is this so? I suppose that $H$ is an infinite dimensional $\Bbb C$ vector space.
I know any compact operator is the limits (in operator norm) of finite rank operator. But this still doesn't really explain the direct limit...
Also why are we only considering matrices of size $2^n$?