# The Trace Class Operators Form a Banach Space

I want examining the trace class operators $L_1(H)$ of a separable Hilbert space $H$ with the norm $||A||_1=\sum\limits^{\infty}_{n=1}\lambda_n$ where $\lambda_n$ are the eigenvalues of $(A^*A)^{1/2}.$ A text I am reading says that this space $L_1(H)$ is a Banach space with respect to that norm. It does not provide any proof though so I tried to but I am having difficulty. I would appreciate if someone could either link me a proof or provide one here. Thanks!

** Someone has posted below an answer using dual spaces. I am still curious though if there is another way to do it more traditionally. **

## 1 Answer

You can prove that $L_1(H)$ is isomorphic to the dual of $K(H)$, the space of compact operators via the map $$A \mapsto \text{tr}(A\cdot \ )$$ which implies that it is a Banach space. The proof should be available in most books on Operator theory - for instance, Murphy's book C* algebras and Operator Theory has a proof (See Section 4.2)

• Is there an easy way to do it by showing a Cauchy sequence converges or absolute converge series implies the operator sum converges? Or is this the only good way to do it? – Leo Spencer Jan 25 '14 at 16:09
• Could you explain why the map you provide is an isometry? I looked at that book and couldn't quite follow as it was on a higher level than I am. – Leo Spencer Jan 25 '14 at 21:17