Mathematics / advanced
Banach spaces and duality
Extend finite-dimensional intuition through completeness, weak topologies, compactness, duality, and operator structure.
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Inside this module
Module 01 advanced
Banach spaces and duality
Extend finite-dimensional intuition through completeness, weak topologies, compactness, duality, and operator structure.
- 01
Normed spaces, completions and quotients
Track the chosen norm through completion and quotient constructions, including the closedness needed for a genuine quotient norm.
29 min - 02
Hahn–Banach extension and separation
Extend scalar functionals without increasing their norm and state real, complex and geometric versions with their proper hypotheses.
31 min - 03
The uniform boundedness principle
Use completeness to turn pointwise bounds into one operator-norm bound, and see exactly how the conclusion fails on an incomplete domain.
30 min - 04
Open mapping and closed graph theorems
Use Banach-space completeness to turn surjectivity or graph closedness into quantitative continuity statements.
31 min - 05
Dual spaces and weak topologies
Identify which functionals test convergence and distinguish weak-star limits from genuinely weak limits on a dual space.
32 min - 06
Banach–Alaoglu and weak-star compactness
Recover compactness in a weaker dual topology while keeping nets, sequences and separable preduals distinct.
34 min - 07
Reflexivity and weak sequential compactness
Use the canonical bidual map and Eberlein–Šmulian to extract weak subsequences without assuming a separable ambient space.
32 min - 08
Compact operators and approximation properties
Separate compactness from finite rank and state precisely when compact maps admit finite-rank approximation in operator norm.
33 min - 09
Schauder bases and unconditionality
Distinguish ordered norm expansions from rearrangement-stable expansions and compute the conditional behavior of the summing basis.
34 min - 10
Interpolation of Banach spaces
Build intermediate spaces from compatible endpoints and transfer operator bounds without confusing interpolation estimates with exact norms.
35 min