By Joseph J. Rotman
A person who has studied summary algebra and linear algebra as an undergraduate can comprehend this publication. the 1st six chapters supply fabric for a primary path, whereas the remainder of the ebook covers extra complex themes. This revised variation keeps the readability of presentation that was once the hallmark of the former variations. From the experiences: "Rotman has given us a truly readable and beneficial textual content, and has proven us many appealing vistas alongside his selected route." --MATHEMATICAL studies
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Extra info for An Introduction to the Theory of Groups, 4th Edition
All linear subspaces of E of a certain fixed finite dimension n, where 1 ^ n < dim E, are semi-Chebyshev (or, what is equivalent, Chebyshev) subspaces. 3°. All closed linear subspaces of E of a certain fixed finite codimension m, where 1 ^ m < dim E, are semi-Chebyshev subspaces. 4°. E is strictly convex. 14, we obtain the next theorem (see [168, p. 111]). 17. For a Banach space E the following statements are equivalent: 1°. All closed linear subspaces ofE are Chebyshev subspaces. 2°. All closed linear subspaces of E of a certain fixed finite codimension m, where 1 ^ m < dim E, are Chebyshev subspaces.
It is well known (see [168, p. 115]) that the space E = c 0 has no Chebyshev subspace of infinite dimension, but it has Chebyshev subspaces of any finite dimension. It is also known that E = L^([0,1], v), where v is the Lebesgue measure, has no Chebyshev subspace of finite dimension or of finite codimension, but still it has Chebyshev subspaces (see below). 1 by combining in some way the spaces r 0 and L^([0, 1], v). 1 only among those separable Banach spaces which are not isometric to any conjugate Banach space (the spaces c0 and LK([O, 1], v) do have this property; moreover, they are even not isomorphic to any conjugate Banach space).
I) IfQ is a compact space, then E = C(Q) has property proxbid. (ii) // T is a locally compact space, then E = C0(T) has property proxbid. (iii) If(T, v) is a positive measure space, then E = L1R(T, v) has property proxbid. , for each e > 0 there exists a compact subset Q of T such that \x(t)\ < e for all t £ T\Q), endowed with the usual vector operations and with the norm ||x|| = sup reT |x(r)|. 16 has been proved by J. Blatter and G. Seever  and, independently, with a different method, by R.