# topological linear space

## topological linear space

[¦täp·ə¦läj·ə·kəl ′lin·ē·ər ‚spās]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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A topological algebra is a topological linear space over the field K (where K stands for either R or C), in which there is defined a separately continuous associative multiplication.
A topological algebra A is a topological linear space over the field K (where K is R or C) with an associative separately continuous multiplication that turns A into an algebra over K.
When the underlying topological linear space of a topological algebra A is locally pseudoconvex, then A is called a locally pseudoconvex algebra.
Let K denote either the field R of real numbers or the field C of complex numbers, X a topological space and Y a topological linear space over K (shortly, a topological linear space), C(X,Y) the set of all continuous maps from X to Y, and [C.sub.0](X,Y) the subset of all such f [omega] C(X,Y) that vanish at infinity.
In , some similar results (Theorem 1 on page 98, Corollaries 1 and 2 on page 99) are presented for a compact Hausdorff space X and a topological linear space Y.
Let Y be a topological linear space and K a subset of Y.
It is said that a topological linear space Y is Klee admissible if for every compact set K [[subset].bar] Y and for every neighbourhood O of zero in Y there exists a continuous finite-dimensional map L:K [right arrow]Y such that L(y)-y[omega] O for every y[omega]K.
It is easy to see that every topological linear space that has the approximation property has also the nonlinear approximation property, and that every topological space that has the nonlinear approximation property is Klee admissible.
826, the authors claim that every locally convex space is Klee admissible and pose an open problem to find out whether every topological linear space is Klee admissible.
Every Hausdorff topological linear space has continuous coordinate functions.
His topics are linear spaces, the algebra of convex sets, topology, metric space topologies, topological linear spaces, measurable spaces and measures, integration, Banach spaces, the differentiability of functions defined on normed spaces, Hilbert spaces, convex functions, optimization, iterative algorithms, neural networks, regression, and support vector machines.
Chapters cover linear spaces and operators, normed linear spaces, major Banach space theorems, Hilbert spaces, Hahn-Banach theorem, duality, topological linear spaces, the spectrum, compact operators, application to integral and differential equations, and spectral theorem for bounded self-adjoint operators.

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