Basic Analysis – Gently Done Topological Vector Spaces

5. Vector Spaces
We shall collect together here some of the basic algebraic results we will need
concerning vector spaces. Wewish toconsider vector spaces either over R, thefield
of real numbers, or over C, the complex numbers, so for notational convenience,
we use the symbol K to stand for either R or C. It is often immaterial which of
these fields of scalars is used, but when it is important we will indicate explicitly
which is meant.
Definition 5.1 A finite set of elements x 1 ,...,x n in a vector space X over K is
said to be linearly independent if and only if
α 1 x 1 +···+α n x n = 0
with α 1 ,...,α n ∈ K implies that α 1 = ··· = α n = 0. A subset A in a vector space
is said to be linearly independent if and only if each finite subset of A is.
Definition 5.2 A linearly independent subset A in a vector space X is called a
Hamel basis of X if and only if any non-zero element x ∈ X can be written as
x = α 1 u 1 +···+α m u m
for some m ∈ N, non-zero α 1 ,...,α m ∈ K and distinct elements u 1 ,...,u m ∈ A.
In other words, A is a Hamel basis of X if it is linearly independent and if any
element of X can be written as a finite linear combination of elements of A.
Note that if A is a linearly independent subset of X and if x ∈ X can be
written as x = α 1 u 1 +···+α m u m , as above, then this representation is unique.
To see this, suppose that we also have that x = β 1 v 1 + ···+β k v k , for non-zero
β 1 ,...,β k ∈ C and distinct elements v 1 ,...,v k ∈ A. Taking the difference, we get
0 = α 1 u 1 +···+α m u m −β 1 v 1 −···−β k v k .
Suppose that m ≤ k. Now v 1 is not equal to any of the other v j ’s and so,
by independence, cannot also be different from all the u i ’s. In other words, v 1
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