Firstly, do we have a definition for the inverse of an infinite matrix? If we have, secondly, how to calculate it? I guess we may approximate it by inverses of finite matrices whose size is increasing larger and larger. But it's just my guess, I can't give a thorough analysis on this approximation, and I think there may be methods that can get the inverse directly, just like finite matrices. I have no idea on this problem, could you please give me a detailed exposition and analysis on this problem? Any suggestion or books on this subject is welcomed. Thanks!