👉 Hilbert's theorem states that every finite-dimensional vector space over a field of characteristic zero has a basis. This means that any finite-dimensional vector space can be written as a direct sum of its irreducible components, which are subspaces with only one non-zero dimension. The proof of this theorem is based on the fact that all vectors in a finite-dimensional vector space can be represented by a finite sequence of coordinates, and it follows that every finite-dimensional vector space has a basis if and only if
