# Companion to Real Analysis by John M. Erdman

By John M. Erdman

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In these notes the only vector spaces which occur are the ones with real or complex scalars. Since the real vector spaces occur most frequently we make the following agreement. 2. Convention. In the sequel the unmodified term “vector space” will mean “real vector space”. When we want a vector space V to have complex scalars, we will say so: we will say Let V be a complex vector space. 3. Example. The vector space V = {0} comprising a single element is the zero vector space. When we say that a vector space is nontrivial we mean that it is not the zero vector space.

In the context of vector spaces, however, only Hamel bases make sense; so the modifier “Hamel” is usually omitted. In a vector space V subsets of linearly independent sets are themselves linearly independent; and supersets of spanning sets are spanning sets. In order for a linearly independent set to be a basis for V it must be “large enough” to span V ; and in order for a spanning set to be a basis for V it must be “small enough” to be linearly independent. Thus in order for a set B to be a basis for V it must be “just the right size”.

Example. Let S be a set which contains at least two elements. Then set inclusion is a partial ordering on the power set P(S) which is not a linear ordering. 8. Example. If S is a nonempty set, then F(S) = F(S, R) can be given a pointwise ordering by setting f ≤ g if f (s) ≤ g(s) for every s ∈ S. Under this ordering F(S) is a partially ordered set. If S contains more than one point, the ordering is not linear. 33 34 5. 2. 1. Definition. Let ≤ be a partial ordering on a set S and A ⊆ S. An element l ∈ S is a lower bound for A if l ≤ a for all a ∈ A.

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