Which classical dimensional formulas in Linear Algebra extend to the infinite-dimensional case?

Question

In Linear Algebra, many formulas related to the dimensions of vector spaces and subspaces are fundamental in the finite-dimensional case. For example, the Rank-Nullity Theorem states that for a linear map $ f: U \to V $, the dimension of the domain satisfies:

$$ \dim(U) = \dim(\ker(f)) + \dim(\operatorname{Im}(f)). $$

I know that this result, along with some others, extends to the infinite-dimensional case, but I would like to understand which other classical dimensional formulas remain valid for infinite-dimensional vector spaces, either from a purely algebraic perspective (based on Hamel bases and Zorn's lemma) or with additional topological structures (e.g., Banach or Hilbert spaces).

For instance:

What happens to the relationship between the dimension of a subspace and its complement in the infinite-dimensional case?
Can results about sums and intersections of subspaces (like $\dim(U + W) + \dim(U \cap W) = \dim(U) + \dim(W)$) be extended?
Are there important limitations or subtleties when dealing with infinite dimensions?

I would highly appreciate detailed explanations, references, or counterexamples that highlight the differences between the finite and infinite-dimensional cases.

I believe such a list would be of great help to any student of Functional Analysis, allowing them to better understand the dimensional tools at their disposal, which are often taken for granted when studying this subject at university.

Thank you in advance!

Here is a reference for your first claim. And the second question is answered in this duplicate. Also this can be generalised to the infinite-dimensional case. — Dietrich Burde, Dec 28 '24 at 11:02
On the other hand, this post lists examples, which do not generalise, like the trace or determinant, or the isomorphism $V\cong V^*$. But this is well known, I suppose. — Dietrich Burde, Dec 28 '24 at 11:08
Other than Grassman's formula (which is true for infinite dimensions as well), what "results about sums and intersections of subspaces" are you referring too? — Sassatelli Giulio, Dec 28 '24 at 11:11

score -1 · Answer 1 · answered Dec 29 '24 at 00:38

You specifically asked about formulas that say sum of dimensions two specific subspaces being equal to the sum of dimensions two other related vector subspaces. As dimensions could be infinite arithmetic of inifinities can be simple in some cases. Suppose $V$ is infinite dimensional and $V$ is finite dimensional, say 6, then the equation in your question becomes something like $\infty + 3 = \infty + 5$ (3 could be any nonnegative integer upto 5)

So my guess is that if both $V,W$ are infinite with one ofthem a "higher infinity" then this equation should be true.

However if the dimensions of both $V$ and $W$ are "same size" infinities this equation may not hold.

Which classical dimensional formulas in Linear Algebra extend to the infinite-dimensional case?

1 Answers1