What is the exact technical difference between a linear operator and linear function?
Wikipedia gives a good description of the similarity between the two:
To me, they are the same thing.
I use linear function, linear operator, linear map, and linear transformation all interchangeably, though you come to notice that different communities of mathematicians have different preferred terminology.
Though I do not believe this is standard if IIRC, Axler defines linear maps (functions) as maps between arbitrary vector spaces but reserves the term operator for maps between the same vector space.
Most use the terms interchangeably in my experience.
I think linear operators must be square matrices will linear maps can be any sort of configuration (i.e. non-square).
That's how linear operators were distinguished from linear transformations when I learned about them. I.e., a linear operator maps a space to itself, hence the matrix for the operator is necessarily square.
I've never see "linear transform" used with infinite dimensional spaces. It always seems to be "operator theory" or "bounded linear operator" or "closed operator".
Linear transform seems to be more of a term you see in linear algebra rather than linear analysis.
Can we say that linear operator and linear function are generally used as synonyms but to be more precise and technically linear operator denotes square matrix since it maps space to itself whereas linear function denotes a rectangular matrix since it maps a vector of one space to different space.
Is that right?
That's pretty close. I would say it this way: a linear operator maps a vector space to itself, which implies that the matrix is square. A linear function maps an arbitrary vector space to a possibly different vector space, which implies that the matrix is not square if the spaces are of different dimension.
Well, you do have Fourier transforms, Gelfand transforms, Laplace transforms, Hilbert transforms,... All of these are on space which are usually pretty infinite-dimensional.
I wouldn't say "more precise". It is simply a word with two different meanings. For some people, it are synonyms, for other people there is a distinction. So it depends on the author.
I would usually use the term endomorphism for a map of a vector space to itself. But of course, other people may use other terminology. No problem with that.
Completely forgot about integral transforms. I was thinking about thinking about things like Open Mapping Theorem :tongue:
Separate names with a comma.