Discussion Overview
The discussion revolves around the question of finding examples of two closed subspaces in a normed or Banach space whose sum is not closed. The focus is primarily on infinite-dimensional spaces, as participants suggest that such examples are difficult to conceive in finite-dimensional settings like R^n.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant suggests that examples of closed subspaces whose sum is not closed likely exist in infinite-dimensional spaces.
- Another participant provides a specific example using the Hilbert space \mathcal{H}=\ell^2(\mathbb{N}), defining a bounded linear operator T and demonstrating that the sum of the subspaces A and B is not closed due to the range of T being a proper dense subspace.
- A third participant shares a link to an external resource that discusses the problem further and mentions relevant literature on the topic.
- A later reply expresses appreciation for the shared resource and acknowledges the usefulness of the search.
Areas of Agreement / Disagreement
Participants do not reach a consensus on a definitive example beyond the one provided, and the discussion remains open to further exploration of the topic.
Contextual Notes
The discussion does not resolve the broader implications of the examples provided, nor does it clarify the conditions under which the sum of closed subspaces may or may not be closed.