Homework Help Overview
The discussion revolves around the properties of two subspaces, S1 and S2, within a Hilbert space defined by an orthonormal basis. The original poster aims to demonstrate that the sum of these subspaces, S1 + S2, is dense in the Hilbert space and to evaluate the density and closedness of S2.
Discussion Character
- Exploratory, Assumption checking, Conceptual clarification
Approaches and Questions Raised
- The original poster considers the closure of S1 + S2 in relation to the orthonormal basis. There is a question about whether an undense set combined with a dense set can result in a dense set. Participants discuss the inclusion of specific basis elements in S1 + S2 and explore the implications for the density and closedness of S2.
Discussion Status
Participants are actively engaging with the problem, questioning assumptions about the relationships between the elements of S1 and S2. There is a recognition of the need for further exploration regarding the properties of S2, with some participants expressing tentative agreement on the conclusions drawn about its density and closedness.
Contextual Notes
There is an ongoing examination of the definitions of dense and closed sets, as well as the implications of the specific vectors involved in the subspaces. The original poster indicates familiarity with these concepts but is seeking clarification on their application in this context.