Discussion Overview
The discussion centers on the mathematical expression involving the supremum of the dot product of two vectors, specifically why the supremum of \( a \cdot u \) subject to the 2-norm of \( u \) is less than or equal to \( r \) times the 2-norm of \( a \). Participants explore the conditions under which this relationship holds and seek clarification on the definition of the 2-norm.
Discussion Character
- Mathematical reasoning
- Conceptual clarification
Main Points Raised
- One participant poses the question regarding the supremum of \( a \cdot u \) under the constraint of the 2-norm of \( u \), asking for a method to work it out.
- Another participant reiterates the same question, indicating a lack of familiarity with the 2-norm and requesting a definition.
- A participant clarifies that the 2-norm refers to the Euclidean norm.
- Further clarification is provided that the supremum can be expressed in conventional notation, suggesting that it holds in finite-dimensional space.
- One participant suggests using the Cauchy-Schwarz inequality to demonstrate the relationship between the left-hand side and the right-hand side of the expression.
Areas of Agreement / Disagreement
Participants express uncertainty regarding the definition of the 2-norm and the mathematical relationship in question. There is no consensus on the resolution of the original query, and multiple viewpoints regarding the understanding of the 2-norm and its implications are present.
Contextual Notes
Some participants have not encountered the 2-norm before, which may affect their understanding of the discussion. The mathematical steps to demonstrate the inequality are not fully resolved.