>>10993451This is my attempt at proving the second one with the "scheme", but I have no idea if thats rigorous enough...

The first one was part of the actual first exam, which I failed. Iirc I tried to prove it by claiming that every differentiable function can be constructed from an (infinite) sum of polynomials, so all the polynomials from zero to infinity form a linear base for the given space, which and are obviously part of, like with a taylor expansion, and that trivially there is no linear operation that changes the degree of a polynomial or something like that, but that wasnt what the prof wanted and he gave me zero score (need like 18 in total to pass so thats pretty significant...)

>>10993468Is it kinda what I did in pic related?

>>10993471This really is my problem. I never got the grasp of learning definitions by memory and then actually applying them, but even now where I have the definitions right in front of me I struggle to see how to apply them. I know that this shit is supposedly easy as fuck and everyone but me seems to be able to pass it, whereas Im now gonna start my bachelor thesis while possibly failing completely cause Im stuck in second semester maths...

Also I dont think is the solution, cause no way anyone can give six points for that and it doesnt use the fact that its differentiable functions and in the exam review we got to talk to the prof and he explained in great detail what I should have done, but that was weeks and a bunch of other exams ago so I dont remember everything.