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Many approximations are linear, that is, conform to the principle of super-position, and may profitably be studied by means of the theory of linear spaces. ``Linear approximation'' sets forth the pertinent parts of that theory, with particular attention to the key spaces $C_n, B, K$, and Hilbert space, spaces which are powerful tools in the analysis, appraisal, and design of approximations, ranging from formulas of mechanical quadrature to approximations of operators by operators. Because it affords a detailed treatment of a timely and important subject, ``Linear approximation'' is of interest to scientists and engineers as well as to mathematicians. The book includes worked illustrative examples and discussions of the rationale of its formulation of problems.