Contents Overview Sets, spaces and sequences Sets and operations Sets Membership and inclusions Set operations Relations Invertibility Basic spaces and their topology Vector spaces Normed spaces Metric spaces Balls and spheres Open and closed sets Limits and completions Limits Completeness Completions Linear spaces and transformations Subspaces Linear dependence Bases Schauder Bases Basis transformations Gaussian elimination Linear transformations Bounded linear transformations Solving differential equations General existence theorems Spectral theory Hilbert space theory Inner-product spaces Orthogonality Adjoints and decompositions