I'm trying to prove that trace class is Banach space under the norm $||\cdot||_{1}$ using the next mehodology:
If $\{T_{n}\}_{n}$ is a Cauchy sequence under $||\cdot||_{1},$ because of the inequality $||\cdot||\leq||\cdot||_{1},$ $\{T_{n}\}_{n}$ is a Cauchy sequence under $||\cdot||.$ Then, there is $T$ linear and bounded on $H,$ complex Hilbert space, such that $T_{n}\rightarrow T.$
The next step is to prove that $Tr(|T|)<\infty.$
To finish this is enough to prove that $$\sum_{\gamma\in\Gamma}|||T|^{1/2}e_{\gamma}||^{2}\leq\limsup_{n}\sum_{\gamma\in\Gamma}|||T_{n}|^{1/2}e_{\gamma}||^{2}$$ because the left side is definition of $Tr(|T|)$ and the right side is bounded by $\sup_{n}||T_{n}||,$ where $\{T_{n}\}$ is Cauchy.
We have the equality $$\sum_{\gamma\in\Gamma}|||T|^{1/2}e_{\gamma}||^{2}=\lim_{n}\sum_{\gamma\in\Gamma}|||T_{n}|^{1/2}e_{\gamma}||^{2}$$ because the continuity of the norm, isn't it? So such inequality has been proved. Is there another way to prove this with the inequality sign?
Then, I'd like to prove that $||T_{n}-T||_{1}=\sum_{\gamma\in\Gamma}|||T_{n}-T|e_{\gamma}||^{2}<\infty$ but I have troubles bounded it. Is there an easy form to bound this?
Any kind of help is thanked in advanced.