In effective field theories we deal with bases of operators. I wonder in which sense is this similar to bases of a vector space. We can change bases and write the Lagrangian in another basis, just as we can do with bases of vectors in a vector space. We also have independent operators, just as in vector spaces.

So, can the concept of bases of opeators be given a vector space structure?

What has triggered this question is me asking myself how can we know when two given operators are independent. I mean, with no criteria (as far as I know) for orthonormality how can we say when two operators are independent?