In this video, I cover three examples of vector spaces. First, we take a look at the set of sequences with entries in a field F. Second, we introduce the function space F^S. Finally, we glimpse the space of polynomials with coefficients in F. The material follows Sheldon Axler's "Linear Algebra Done Right", Chapter 1B. Correction: In the first example, I incorrectly state that I will prove the equality of the sets F^infinity and F^N. These two sets are equal by definition if we define sequences as functions with domain N (as is usual). What I demonstrate, is how this definition translates into the usual notation. If we would define sequences in another way, the same argument would show that F^infinity and F^N are isomorphic.