[Math] Subspaces of an infinite dimensional vector space

vector-spaces

It is well known that all the subspaces of a finite dimensional vector space are finite dimensional.
But it is not true in the case of infinite dimensional vector spaces. For example in the vector space $\mathbb{C}$ over $\mathbb{Q}$, the subspace $\mathbb{R}$ is infinite dimensional, whereas the subspace $\mathbb{Q}$ is of dimension 1.

Now I want examples (if it exist) of the following:

  1. An infinite dimensional vector space all of whose proper subspaces are finite dimensional.

  2. An infinite dimensional vector space all of whose proper subspaces are infinite dimensional.

Best Answer

You won't find anything like that. If $V$ is an infinite dimensional vector spaces, then it has an infinite basis. Any proper subset of that basis spans a proper subspace whose dimension is the cardinality of the subset. So, since an infinite set has both finite and infinite subsets, every infinite dimensional vector space has both finite and infinite proper subspaces.

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