Abstract Linear Spaces

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Abstract Linear Spaces

Abstract linear spaces are also called vector spaces, and they occur in a wide variety of mathematical settings. The most familiar examples are the finite dimensional Euclidean spaces. Let N be a positive integer. Then ℜ^N can be represented as the set of all column vectors

in which each coordinate x_n, 1nN is an element from the field ℜ of real numbers. Sometimes it is more convenient to represent the elements of ℜ^N as row vectors, but for the purposes of this article column vectors will be used. There are two basic algebraic operations that are defined in ℜ^N. The first is addition of vectors. If x and y are two vectors in ℜ^N then x + y is the vector obtained by adding the corresponding coordinate of x and y. Thus if we write the vectors as columns we find that

The second operation is scalar multiplication, that is multiplication of a vector by a scalar. A scalar simply means an element from the field ℜ. If is a real number and x is a vector then x is the vector defined by

Thus x is obtained by multiplying each coordinate of the vector x by the real number . It is easy to verify that the set ℜ^N together with the operation + on vectors is an Abelian group. The zero vector

is the identity element of the group. The operation of scalar multiplication satisfies the identities

0x = 0, 1x = x, ()x = (x),

and also the two distributive laws

( + )x = x + x, and (x + y) = x + y.

the triple consisting of the set ℜ^N and the two operations of addition and scalar multiplication form a linear space or vector space over the field ℜ of real numbers. Note that we have determined a vector space for each positive integer N, but, for example, the spaces ℜ³ and ℜ⁴ are distinct.

Here is another example of a vector space, but in this case the field of scalars is the field C of complex numbers. Let = {z ∈ C: |z|<1} be the open unit disk in C. Then let A() denote the set of all bounded analytic functions ƒ: C. Thus an analytic function ƒ: C belongs to A() if there exists a nonnegative constant C_ƒ such that

|ƒ(z)| C_ƒ

for all complex numbers z in . If ƒ and g belong to A() we define their sum ƒ + g to be the function

(ƒ + g)(z) = ƒ(z) + g(z).

Notice that the + sign on the left defines addition of elements from the set A(), while the + sign on the right is addition of complex numbers. The sum ƒ + g defined in this way is again an analytic function on . If C_ƒ and C_g are nonnegative constants such that

|ƒ(z)| C_ƒ and |g(z)| C_g.

for all complex numbers z in , then, by the triangle inequality for complex numbers, we have

|(ƒ + g)(z)| = ƒ(z) + g(z)| |ƒ(z)| + |g(z)| C_ƒ + C_g.

This shows that ƒ + g is a bounded analytic function and so is an element of A(). In a similar manner we define scalar multiplication of the function ƒ by the complex number by

(ƒ)(z) = ƒ(z).

Then it is obvious that ƒ is a bounded analytic function on . The set A() together with the operations of addition and scalar multiplication satisfy algebraic identities analogous to those satisfied by the vector spaces ℜ^N. In this case, however, A() is a vector space over the field C of complex numbers. In this example we could drop the requirement that the analytic functions ƒ are bounded, and the resulting set would continue to form a vector space over C with the same definitions for addition and scalar multiplication. However, the most important examples of vector spaces often impose such additional restrictions and then, of course, one must verify that the restriction is preserved by the algebraic operations of addition and scalar multiplication. The space A() is typical of this sort of construction.

The concept of an abstract linear space is a generalization of the constructions just described for the spaces ℜ^N and A(). Let F be a field and (V, +) an Abelian group. It will be convenient to write 0F for the additive identity element of F, 1F for the multiplicative identity element of F, and 0 for the additive identity element in the group V. Now assume that there exists a function (, v) v from F x V V, denoted by juxtaposition, which satisfies the following conditions:

0_Fv = 0, 1_Fv = v, ()v = (v),

and the two distributive laws

( + )v = v + v, and (u + v) = u + v.

Here and are elements of the field F, and u and v are elements of the group V. Then V is a vector space or linear space over the field F. The elements of V are called vectors and the elements of F are called scalars.

Notice that the special spaces ℜ^N and A() are examples of vector spaces over the fields of real numbers and complex numbers, respectively. However, the concept of an abstract linear space is much more general and includes, for example, linear spaces for which the field F of scalars is a finite field. Here is an example of a vector space over a finite field. Let p be a prime number and write F_p for the field with p elements. We can use the field of integers modulo p for F_p because every finite field with p elements is isomorphic to the integers modulo p. Then F_p^N can be represented as the set of all column vectors

in which each coordinate a_n, 1 n N, is an element from the finite field F_p. Addition and scalar multiplication in F_p^N are defined by exactly the same formulas that we used in ℜ^N, but now the scalars and the entries in the vectors are elements of F_p^N. It is easy to verify that F_p^N is a vector space over the finite field F_p.

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