SUMMARY
The supremum of the dot product of vectors a and u, constrained by the 2-norm of u, is established as less than or equal to r times the 2-norm of a, formally expressed as sup{a.u | ||u||_2 <= r} = r ||a||_2. This relationship is derived from the Cauchy-Schwarz inequality, which provides a foundational proof for this inequality in finite-dimensional spaces. The 2-norm, also known as the Euclidean norm, is crucial for understanding this concept.
PREREQUISITES
- Understanding of vector spaces and dot products
- Familiarity with the Cauchy-Schwarz inequality
- Knowledge of the 2-norm (Euclidean norm) in finite-dimensional spaces
- Basic concepts of supremum in mathematical analysis
NEXT STEPS
- Study the Cauchy-Schwarz inequality in detail
- Explore the properties and applications of the 2-norm in vector analysis
- Learn about supremum and infimum in mathematical contexts
- Investigate other norms such as 1-norm and infinity norm for comparative analysis
USEFUL FOR
Mathematicians, data scientists, and students in linear algebra or optimization who seek to deepen their understanding of vector norms and inequalities.