Lectures related to Metric Spaces: Norms and Distances

Covers vector spaces, scalar products, norms, and forms of polarization in standard properties.

Introduces metric spaces, topology, and continuity, emphasizing the importance of open sets and the Hausdorff property.

Explores orthogonality, triangle inequality, and the Pythagorean theorem in vector spaces.

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.

Covers the definition of norm, distance between vectors, and Cauchy-Schwarz inequality.

Explores the scalar product, vector lengths, angles, norms, and rotations in 2D and 3D spaces.

Covers the properties of Euclidean spaces, focusing on R^n and its applications in analysis.

Explores orthogonal matrices and transformations, emphasizing preservation of norms and angles.

Covers normed vector spaces, topology in R^n, and the principle of drawers as a demonstration method.

Explores orthogonality, scalar product, and orthonormal bases in vector spaces.

Covers the properties of vectors, including commutativity, distributivity, and linearity.

Introduces the basics of real vector spaces, norms, and scalar products.

Covers linear applications, diagonalizable matrices, eigenvectors, and orthogonal subspaces in R^n.

Covers convergence, completeness, and properties of metric spaces and Banach spaces.

Explores the properties of orthonormal vectors in Euclidean space through key equations and demonstrations.

Covers the construction of Qp, norms, valuations, and Ostrovski's theorem.

Explores orthogonality, dot product properties, vector norms, and angle definitions in vector spaces.

Covers the definition of scalar product, properties, examples, and applications in Euclidean spaces, including the Cauchy-Schwartz inequality.

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

Explores optimal transport in heat equations and metric spaces.