Normed vector spaces
Objective. Compare norms and analyse convergence.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A normed vector space is a vector space V equipped with a norm ‖·‖ satisfying ‖v‖≥0 with equality iff v=0, ‖cv‖=|c|‖v‖, and the triangle inequality ‖u+v‖≤‖u‖+‖v‖; different norms on the same space can induce different notions of convergence, though on finite-dimensional spaces all norms are equivalent. On ℝⁿ the common norms are ‖x‖₁=Σ|xᵢ|, ‖x‖₂=√(Σxᵢ²), and ‖x‖_∞=max|xᵢ|, and a sequence converges in one norm iff it converges in any equivalent norm on finite dimensions. For example, in infinite-dimensional function spaces like C[0,1], the sup norm ‖f‖_∞ and the L¹ norm ‖f‖₁=∫|f| are not equivalent, so convergence in one does not guarantee convergence in the other.
Worked example
Compute the ‖·‖_∞ norm of the vector v = (3, −5, 2) in ℝ³ using the normed space definition.
- 1‖v‖_∞ is defined as the maximum of the absolute values of the components
- 2Compute |3|=3, |−5|=5, |2|=2
- 3Take the maximum of these three values
- 4‖v‖_∞ = 5
Answer: 5
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Practice
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Start practisingCommon mistake. Students assume all norms on any vector space are equivalent, which only holds in finite dimensions; in infinite-dimensional spaces, inequivalent norms can give genuinely different convergent sequences, so always check dimensionality before invoking equivalence.
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