**Background**  

This is a follow-up question to:
[What (classes of) Banach spaces are known to have Schauder basis?](https://mathoverflow.net/questions/134727/what-classes-of-banach-spaces-are-known-to-have-schauder-basis)


In the previous question, I asked about what spaces are known to have Schauder basis. It seems that not a lot of positive results are available in that area. So I am restricting my question to one of the spaces that is in particular relevant to my research:



> 
> Does $C\_p$, the Banach space of all Schatten-p operators on a separable infinite-dimensional Hilbert space, have property $\pi$? Or does it have a (Schauder) basis?
> 
> 
> 


Thank you!


**Reference**  

Good reference for different approximation properties: *Handbook of the Geometry of Banach Spaces, vol. 1 -- contribution by Pete Casazza*



What Approximation Property does the space of Schatten-p class operators have?
It has a Schauder basis, namely $e\_i\otimes e\_j$, where $(e\_i)$ is an orthonormal basis of the Hilbert space. This holds for $1\le p<\infty$. For the Schatten ideal of compact operators, I do not know the answer. 


