∞

Measure Theory & Functional Analysis

Rigorous foundations: Lebesgue integration, Hilbert spaces, and RKHS for kernel methods.

6 concepts

Intermediate1

Universal Approximation Theorems

The Universal Approximation Theorems say that a neural network with at least one hidden layer and a suitable activation can approximate any continuous function on a compact domain as closely as you like.

#universal approximation theorem#cybenko#hornik+12

Advanced5

∑MathAdvanced

Sigma-Algebras & Measure Spaces

A σ-algebra is a collection of subsets that is closed under complements and countable unions, giving us a stable universe of sets where measure makes sense.

#sigma-algebra#measure space#measurable sets+12

∑MathAdvanced

Lebesgue Integration

Lebesgue integration measures how much time a function spends near each value and adds up value × size of the set where it occurs.

#lebesgue integral#riemann integral#measure theory+12

∑MathAdvanced

Hilbert Spaces

A Hilbert space is an inner product space that is complete, meaning Cauchy sequences converge to points inside the space.

#hilbert space#inner product#l2 space+12

∑MathAdvanced

Banach Spaces

A Banach space is a vector space with a norm where every Cauchy sequence actually converges within the space.

#banach space#normed vector space#completeness+11

📚TheoryAdvanced

Reproducing Kernel Hilbert Spaces (RKHS)

An RKHS is a space of functions where evaluating a function at a point equals taking an inner product with a kernel section, which enables the “kernel trick.”

#rkhs#kernel trick#gram matrix+12