Abstract
When the vector spaces ℝ2 and ℝ3 are pictured in the usual way, we have the idea of the length of a vector in ℝ2 or ℝ3 associated with each vector. This is clearly a bonus which gives us a deeper understanding of these vector spaces. When we turn to other (possibly infinite-dimensional) vector spaces, we might hope to get more insight into these spaces if there is some way of assigning something similar to the length of a vector for each vector in the space. Accordingly we look for a set of axioms which is satisfied by the length of a vector in ℝ2 and ℝ3. This set of axioms will define the “norm” of a vector and throughout this book we will mainly consider normed vector spaces. In this chapter we investigate the elementary properties of normed vector spaces.KeywordsBanach SpaceVector SpaceNormed SpaceSpace PropertyLinear SubspaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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