Functional Analysis

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Practice Lab

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Normed Spaces - Exercises

8 exercises covering norm axioms, non-norm counterexamples, unit balls, norm equivalence, completeness, operator norms, dual norms, and contraction mappings.

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Difficulty Levels

Level	Exercises	Focus
Easy	1-3	Core definitions and geometry
Medium	4-6	Convergence and operators
Hard	7-8	Duality and fixed points

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Exercise 1 (Easy) - Verify Norm Axioms

Check positivity, homogeneity, and triangle inequality for common norms.

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Exercise 2 (Easy) - Find Non-Norm Counterexamples

Show why $\ell^0$ and $p<1$ quasi-norms fail.

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Exercise 3 (Easy) - Unit Ball Geometry

Interpret $\ell^1$ , $\ell^2$ , and $\ell^\infty$ unit balls.

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Exercise 4 (Medium) - Norm Equivalence Constants

Verify finite-dimensional norm inequalities.

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Exercise 5 (Medium) - Cauchy and Completeness

Study rational approximations to an irrational limit.

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Exercise 6 (Medium) - Operator Norm

Compute and sample-check a matrix spectral norm.

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Exercise 7 (Hard) - Dual Norms and Holder

Verify dual norm inequalities numerically.

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Exercise 8 (Hard) - Contraction Mapping

Implement a contraction and find its fixed point.

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Exercise 9: Lipschitz Constants Across Norms

Estimate the operator size of a matrix under several vector norms, then explain why the chosen norm changes the certified Lipschitz constant.

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Exercise 10: Contraction Rate in a Complete Normed Space

Iterate an affine contraction and estimate the observed error ratio. Connect the numerical behavior to the Banach fixed-point theorem.

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