I just wanna know if my argument could be written down on my final exam paper as a correct formal argument.
This is the problem: If m<n, show that there is a 1-1 mapping such that for all .
well, my idea is that with every element f in I associate an element f' in defined in this way:
If then , otherwise: .
then I define as . It's easy to show that this operation is well defined. Now, I define the map by .
To see that this map is one-to-one, suppose . from the definition of , It's easy to see that .
Now, with the definition of the function F, It's clear that and we're done.