Normed spaces
Measure size and distance in vector spaces while distinguishing equivalent and nonequivalent notions of norm.
Side 216Side Studies / Research
Subject
Functional Analysis
Purpose
Infinite-dimensional vector spaces and operators studied through norms, inner products, completeness and spectral structure.
Structure
05 movesFormal systemV0
Definitions → structures → relations → proof → application
Extend linear algebra into spaces of functions.
Functional analysis provides the language for differential equations, quantum theory and optimization where the unknown is often a function rather than a finite vector.
Banach spaces
Use completeness to ensure convergent sequences remain inside the space under analysis.
Hilbert spaces
Add inner-product geometry, orthogonality and projection to infinite-dimensional settings.
Linear operators
Study boundedness, continuity, kernels, ranges and compactness for transformations between function spaces.
Spectral theory
Generalize eigenvalues and decompositions to operators governing differential equations and quantum systems.
Keep the boundaries visible.
function space ≠ finite vector space
bounded operator ≠ finite-valued output
complete ≠ compact
Questions that organize the Side.
Why is completeness essential in existence proofs?
Which finite-dimensional intuitions fail in infinite-dimensional spaces?
How does spectral structure connect operators to solvable modes?
What should carry weight here?
Arguments should state the topology, norm and domain of operators explicitly; finite-dimensional analogies are guides, not proofs.