Competencies: Mathematics > Analysis

Analysis

Covers Real Analysis, Complex Analysis, Functional Analysis, and Measure Theory as one unified competency.

Body of Knowledge

Topic	Description	Relevance	Career Tracks
Sequences & Series Convergence	Epsilon-N definition, Cauchy sequences, absolute vs. conditional convergence, convergence tests (ratio, root, comparison, integral).	High	ML Engineer, Quantitative Analyst, Physicist
Metric Spaces	Distance functions, open balls, convergence in metric spaces, examples (Euclidean, discrete, function spaces), equivalent metrics.	High	ML Engineer, Research Mathematician
Completeness	Complete metric spaces, Banach fixed-point theorem, completion of metric spaces, R as completion of Q.	High	ML Engineer, Quantitative Analyst, Research Mathematician
Compactness in Analysis	Sequential compactness, Heine-Borel theorem, Bolzano-Weierstrass, Arzela-Ascoli theorem, compact operators.	High	ML Engineer, Quantitative Analyst
Complex Differentiability	Holomorphic functions, Cauchy-Riemann equations, power series representation, analytic continuation, isolated singularities.	High	Physicist, Quantitative Analyst, ML Engineer
Cauchy Integral Formula	Contour integration, Cauchy’s theorem, integral formula for derivatives, Liouville’s theorem, fundamental theorem of algebra.	High	Physicist, Quantitative Analyst
Residue Theorem	Laurent series, classification of singularities, residue computation, evaluation of real integrals via contour methods.	High	Physicist, Quantitative Analyst, ML Engineer
Conformal Maps	Angle-preserving maps, Riemann mapping theorem, Mobius transformations, applications to fluid dynamics and electrostatics.	Medium	Physicist, Quantitative Analyst
Lebesgue Measure & Integration	Sigma-algebras, measurable sets, Lebesgue measure, measurable functions, Lebesgue integral, dominated convergence theorem, Fubini’s theorem.	Critical	ML Engineer, Quantitative Analyst, Physicist
L^p Spaces	p-norms on function spaces, Holder and Minkowski inequalities, completeness (Banach spaces), dual spaces, L^2 as Hilbert space.	High	ML Engineer, Quantitative Analyst, Physicist
Banach & Hilbert Spaces	Complete normed and inner product spaces, orthogonality, projections, Riesz representation theorem, separability.	High	ML Engineer, Physicist, Quantitative Analyst
Bounded Operators	Operator norm, bounded linear operators between Banach spaces, dual spaces, Hahn-Banach theorem, open mapping theorem.	Medium	Physicist, Quantitative Analyst, Research Mathematician
Spectral Theory	Spectrum of an operator, eigenvalues in infinite dimensions, spectral theorem for compact self-adjoint operators, applications to quantum mechanics.	High	Physicist, Quantitative Analyst, ML Engineer

Topic

Description

Relevance

Career Tracks

Sequences & Series Convergence

Epsilon-N definition, Cauchy sequences, absolute vs. conditional convergence, convergence tests (ratio, root, comparison, integral).

High

ML Engineer, Quantitative Analyst, Physicist

Metric Spaces

Distance functions, open balls, convergence in metric spaces, examples (Euclidean, discrete, function spaces), equivalent metrics.

High

ML Engineer, Research Mathematician

Completeness

Complete metric spaces, Banach fixed-point theorem, completion of metric spaces, R as completion of Q.

High

ML Engineer, Quantitative Analyst, Research Mathematician

Compactness in Analysis

Sequential compactness, Heine-Borel theorem, Bolzano-Weierstrass, Arzela-Ascoli theorem, compact operators.

High

ML Engineer, Quantitative Analyst

Complex Differentiability

Holomorphic functions, Cauchy-Riemann equations, power series representation, analytic continuation, isolated singularities.

High

Physicist, Quantitative Analyst, ML Engineer

Cauchy Integral Formula

Contour integration, Cauchy’s theorem, integral formula for derivatives, Liouville’s theorem, fundamental theorem of algebra.

High

Physicist, Quantitative Analyst

Residue Theorem

Laurent series, classification of singularities, residue computation, evaluation of real integrals via contour methods.

High

Physicist, Quantitative Analyst, ML Engineer

Conformal Maps

Angle-preserving maps, Riemann mapping theorem, Mobius transformations, applications to fluid dynamics and electrostatics.

Medium

Physicist, Quantitative Analyst

Lebesgue Measure & Integration

Sigma-algebras, measurable sets, Lebesgue measure, measurable functions, Lebesgue integral, dominated convergence theorem, Fubini’s theorem.

Critical

ML Engineer, Quantitative Analyst, Physicist

L^p Spaces

p-norms on function spaces, Holder and Minkowski inequalities, completeness (Banach spaces), dual spaces, L^2 as Hilbert space.

High

ML Engineer, Quantitative Analyst, Physicist

Banach & Hilbert Spaces

Complete normed and inner product spaces, orthogonality, projections, Riesz representation theorem, separability.

High

ML Engineer, Physicist, Quantitative Analyst

Bounded Operators

Operator norm, bounded linear operators between Banach spaces, dual spaces, Hahn-Banach theorem, open mapping theorem.

Medium

Physicist, Quantitative Analyst, Research Mathematician

Spectral Theory

Spectrum of an operator, eigenvalues in infinite dimensions, spectral theorem for compact self-adjoint operators, applications to quantum mechanics.

High

Physicist, Quantitative Analyst, ML Engineer

Personal Status

To be populated after initial study and self-assessment.