What are natural examples of non-relativizable proofs?

What are the most classic examples of non-linear spaces?

• While working on one of the proofs on normed spaces I found that linearity was assumed with a statement that almost all spaces we shall consider will be linear. I wonder now that what are the spaces which are non linear? What would be a classical example of such a space? Which topic of Applied Mathematics makes use of such spaces? Any recommendations of a good book on an introduction will be highly appreciated.

• Answer:

  In this context, "linear space" is a somewhat older term for what we now call a vector space.  It's easy to come up with examples of things that aren't vector spaces: for instance, {(x,x2):x∈R}{(x,x2):x∈R}\{(x, x^2) : x \in \mathbb{R}\}.