Give a nontrivial example of an infinite dimensional vector

Give a nontrivial example of an infinite dimensional vector space that contains a finite dimensional subspace (your subspace can’t just be the zero vector). Be sure to prove that the subspace is a subspace and has finite dimension.

Solution

An affine space over an infinite field F is not the union of n proper affine subspaces.

The inductive step goes like this: Pick one of the affine subspaces V. Pick an affine subspace of codimension one which contains it, W. Look at all the translates of W. Since F is infinite, some translate W of W is not on your list. Now restrict all other subspaces down to W and apply the inductive hypothesis.

This gives the tight bound that an F affine space is not the union of n proper subspaces if |F|>n. For vector spaces, one can get the tight bound |F|n by doing the first step and then applying the affine bound.