TITLE: Isomorphism between infinite dimensional vector spaces
QUESTION [9 upvotes]: Does defining an isomorphism $\theta: \mathbb R^{\mathbb N} \to \{\text{polynomials}\}$ make sense? It does intuitively, but I am worried about the infinite nature. Thanks.

REPLY [3 votes]: Think about these:

the set of sequences of real numbers of length $n$
the set of finitely long sequences of real numbers
the set of infinite sequences of real numbers such the sum of their squares is finite
the set of all infinitely long sequences of real numbers

The first is a finite-dimensional vector space.  The next three are infinite-dimensional vector spaces.  They're not all the same space.  When you understand the difference between them, you're on your way to understanding infinite-dimensional vector spaces.