Homework Help Overview
The discussion revolves around the properties of self-adjoint operators in the context of linear operators on inner product spaces. The original poster is attempting to prove that if a linear operator T is self-adjoint on an inner product space V, then its restriction Tw to a T-invariant subspace W is also self-adjoint.
Discussion Character
- Conceptual clarification, Assumption checking, Mathematical reasoning
Approaches and Questions Raised
- The original poster attempts to extend a basis from the subspace W to the entire space V, suggesting that if T is self-adjoint, then the basis for W can also be orthonormal and consist of eigenvectors of T.
- Some participants question the validity of the original proof, particularly regarding the selection of an orthonormal basis for W and the implications of self-adjointness.
- Others seek clarification on the definition of self-adjoint operators and how it applies to the restriction of T to W, raising concerns about the immediate nature of the proof.
- There is a discussion about the relationship between the bases of the subspace and the containing space, with one participant providing a counterexample to illustrate potential misconceptions.
Discussion Status
The discussion is active, with participants exploring various interpretations of the proof and the properties of self-adjoint operators. Some guidance has been offered regarding the definitions involved, but there is no explicit consensus on the original poster's proof or its implications.
Contextual Notes
Participants are navigating assumptions about the nature of bases in subspaces versus their containing spaces, as well as the implications of self-adjointness in this context. The discussion reflects a need for clarity on these foundational concepts.