By M.A. Armstrong

ISBN-10: 0387908390

ISBN-13: 9780387908397

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2 is concerned with establishing the notion of dimension for an arbitrary linear space. In particular, the dimension need not be finite, which, at times, necessitates some more intricate proofs. 3 discusses linear operators, the natural choice of structure preserving functions between linear spaces, studies their basic properties, and discusses the notion of isomorphic linear spaces. 4 introduces standard constructions producing new spaces from given ones, and in particular the kernel and image of a linear operator are discussed.

Xm , 0, 0, 0, . ) | m ≥ 1, x1 , . . , xm ∈ C} of all sequences that are eventually 0. 3, each of these sets is easily seen to be a linear space over C. Obviously, replacing C throughout by R yields similar linear spaces over R. 5 The constructions given above of the linear spaces Rn , Cn , R∞ , and C∞ easily generalize to any field K and to any cardinality. Indeed, consider an arbitrary set B and an arbitrary field K . Recall from the Preliminaries (Sect. 8) that the set K B is the set of all functions x : B → K .

But that would imply that R is a countable set while the reals are well-known to be uncountable. We conclude that R, as a linear space over Q, is infinite dimensional of uncountable dimension. We close this section by illustrating a difference between finite dimensional linear spaces and infinite dimensional ones. 4 The cardinality of any linearly independent set A in a linear space V is a lower bound for the dimension of V . Moreover, if V is finite dimensional, then any linearly independent set A ⊆ V whose cardinality is equal to the dimension of V is a basis.

### Basic topology by M.A. Armstrong

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