$\mathcal{M}_{n}(\mathbb{C})$ as a Hilbert space

Question

Let $\mathcal{M}_{n}(\mathbb{C})$ be the set of $n\times n$ matrices over $\mathbb{C}.$ I know that for $A,B\in \mathcal{M}_{n}(\mathbb{C}),$ $$\langle A,B \rangle=\text{tr}(B^{*}A)$$ defines an inner product on $\mathcal{M}_{n}(\mathbb{C})$ and hence we can induce the norm.

I have a problem on how to verify that every Cauchy sequence of $\mathcal{M}_{n}(\mathbb{C})$ converges on $\mathcal{M}_{n}(\mathbb{C})$.

Any hint/help would be appreciated.

Answer 1

The dimension of $M_n(\mathbb{C})$ is finite and in a finite-dimensional space all norms are equivalent. Therefore, you can work as if your space was $\mathbb{C}^{n^2}$, endowed with its usual norm, which is complete.

Answer 2

Since $\mathcal{M}_n(\mathbb{C})$ is finite-dimensional, it is complete. Whence the result.

In fact, $\mathcal{M}_n(\mathbb{C})$ equipped with the given Hermitian product is just $\mathbb{C}^{n^2}$ with its usual Hermitian structure.