Hello,


in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and which behave very similarly. These include


* $\mathbb R^n$ with a $p$-norm
* $l^p$ spaces
* $L^p$ spaces
* $p$-Trace-norms on matrices
* $p$-Schatten-classes


Are you aware of an approach that handles these objects in an unified abstract way? An interesting result would be the construction of such a family of spaces from a given Hilbert space.



General theory for p-normed spaces
As Nate and I pointed out in comments, your question reduces to asking whether there is a unified framework which includes both $L^p$ spaces and Schatten spaces. One such framework (there may be others) is **noncommutative $L^p$ spaces**. (Usual $L^p$ spaces are the commutative special case.) There's a [nice survey article](http://books.google.com/books?id=-2LFp8aQ1_UC&lpg=PP1&ots=nHgjkVUtNj&dq=handbook%2520of%2520the%2520geometry%2520of%2520banach%2520spaces&pg=PA1459#v=onepage&q&f=false) by Pisier and Xu in the *Handbook of the Geometry of Banach Spaces*.


