Hölder's inequality

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Embedding in Function Spaces

Explores embedding in function spaces, including elliptic equations, Hölder inequality, Lorentz space, and Hardy-Young inequality.

Finite Element Method: Weak Solutions

Covers weak solutions in the finite element method, emphasizing continuity and the Cauchy-Schwarz inequality.

Concentration Inequalities: Hoeffding's Inequality

Covers Hoeffding's inequality and concentration inequalities with a focus on sequences of random variables.

Properties of Complete Spaces

Covers the properties of complete spaces, including completeness, expectations, embeddings, subsets, norms, Holder's inequality, and uniform integrability.

Signals & Systems II: Discrete Signal Spaces

Explores discrete signal spaces, non-Euclidean norms, and the distinction between bounded and unrestricted signals.

Conditional Expectation: Properties & Jensen's Inequality

Covers the properties of conditional expectation and Jensen's inequality in probability theory.

K-means and Gaussian Mixture Model

Introduces K-means clustering, the Gaussian mixture model, Jensen's inequality, and the EM algorithm.

Convex Optimization: Dual Cones

Explores dual cones, generalized inequalities, SDP duality, and KKT conditions in convex optimization.

Convex Sets and Functions

Introduces convex sets and functions, discussing minimizers, optimality conditions, and characterizations, along with examples and key inequalities.

Cauchy-Schwarz Inequality: Proof and Applications

Explores the Cauchy-Schwarz inequality, norms, and inner products in a real inner product space.