2

I am aware a similar question had been answered before on the site, see(A question about complement of a closed subspace of a Banach space). I find the answer given there unsatisfying since it relies on a categorical definition of direct sum where as my definition is strictly algebraic.

In my book the notion of a direct sum simply asserts a unique presentation of any element in $B$ as a sum of an element in $M$ and an element in $N$ (both of which are linear sub-spaces of $B$). In that case is it still true that $N$ has to be closed?

Aweygan
  • 23,232
roy yanai
  • 369
  • 1
  • 8
  • Just to clarify, by $M\oplus N=B$ you mean that $M$ and $N$ are subspaces of $B$, $M+N=B$, and $M\cap N={0}$? – Aweygan Apr 10 '18 at 13:03
  • 1
    If you're using the purely algebraic notion of direct sum, then "no". Let $N$ be any non-closed subspace of finite codimension. But the purely algebraic notion is not well-suited in functional analysis. – Daniel Fischer Apr 10 '18 at 13:04
  • could you give a specific example of a non-closed subspace of finite co-dimension? – roy yanai Apr 12 '18 at 15:27

0 Answers0