What you'll learn

30 lessons in Measure Theory & Functional Analysis

σ-algebras & measuresCarathéodory extensionMeasurable functionsThe Lebesgue integralThe convergence theoremsProduct measures & FubiniRadon–Nikodym & signed measures$L^p$ spaces & dualityHahn–Banach theoremThe Baire trioArzelà–AscoliHilbert spaces & orthonormal basesSpectral theorem for operatorsRiesz representation & measures-as-functionalsOuter measure & non-measurable setsModes of convergence; Egorov & LusinWeak topologies & Banach–AlaogluCompact operators & Fredholm theoryThe Lebesgue differentiation theoremConvolution & approximate identitiesThe Fourier transform & PlancherelBounded operators & adjointsThe Hardy–Littlewood maximal functionDistributions & tempered distributionsSobolev spaces & embeddingsUnbounded self-adjoint operatorsOperator semigroups & Hille–YosidaInterpolation: Riesz–Thorin & MarcinkiewiczHaar measure on groupsC*-algebras & Gelfand–Naimark

How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the moment just before you'd forget it. A short placement check finds what you already know, so you start Measure Theory & Functional Analysis exactly where it's useful.

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