By Leonid P. Lebedev;Iosif I. Vorovich;G.M.L. Gladwell
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Extra resources for Functional analysis : applications in mechanics and inverse problems
If it is clear from the context, the terms real or complex will be omitted. Let us consider some properties of X.
1), this means that elements in are equivalence classes of Cauchy sequences of continuous functions. Remember that is a Cauchy sequence in if and that two sequences and are equivalent if 52 2. Introduction to Metric Spaces Let us examine elements in First, we say loosely that if then but this is not strictly accurate. The elements of are not functions, but equivalence classes of Cauchy sequences of functions. What we should say strictly is that if then there is an equivalence class which includes, as one of its Cauchy sequences, the sequence We need to label this equivalence class; we could label it or use the same label Let us explore this deeper.
7. 8. 2 A linear space X is called a normed linear space if, for every a norm satisfying N1-N3 is defined. 3 Let X be a normed linear space. Two norms on X are said to be equivalent if there exist positive numbers that and We shall show below that any two norms in are equivalent. 2) satisfies the axioms D1-D4. This shows that a normed linear space is a metric space. 3 does not satisfy this equation. 8. 4 Let X be a normed linear space, and suppose Y is called a subspace of X if it is itself a linear space, i.