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LINEAR SPACES

A linear space X over a field F is a mathematical object in which two operations are defined: addition and multiplication by scalars.

Addition, denoted by +,as in

(1)

is assumed to be commutative,

(2)

associative,

(3)

and to form a group, with the neutral element denoted as 0:

(4)

The inverse of addition is denoted by-:

(5)

The second operation is the multiplication of elements of X by elements k of the field F:

The result of this multiplication is again an element of X. Multiplication by elements of F is assumed to be associative,

(6)

and distributive,

(7)

as well as

(8)

We assume that multiplication by the unit of F, denoted as 1, acts as the identity:

(9)

These are the axioms of linear algebra. From them proceed to draw some deductions.

Set b = 0 in (8). It follows that for all x,

(10)

Set a = 1, b = –1 in (8). Using (9) and (10), we deduce that for all x,

(11)

The finite-dimensional linear spaces are dealt with in courses on linear algebra. In this book the emphasis is on the infinite-dimensional ones—those that are not finitedimensional. The field F will be either the real numbers or the complex numbers Here are some examples.

Example 1. X is the space of all polynomials in a single variable s, with real coefficients, here F =

Example 2. X is the space of all polynomials in N variables s1, … , s N, with real coefficients, here F =

Example 3. G is a domain in the complex plane, and X the space of all functions complex analytic in G, here F = .

Example 4. X = space of all vectors

with infinitely many real components, here F =

Example 5. Q is a Hausdorff space, X the space of all continuous real-valued functions on Q, here F =

Example 6. M is a C∞ differentiable manifold, X = C∞ (M), the space of all differentiable functions on M.

Example 7. Q is a measure space with measure m, X = L1 (Q, m).

Example 8. X = LP(Q,m).

Example 9. X = harmonic functions in the upper half-plane.

Example 10. X = all solutions of a linear partial differential equation in a given domain.

Example 11. All meromorphic functions on a given Riemann surface; F = .

We start the development of the theory by giving the basic constructions and concepts. Given two subsets S and T of a linear space X, we define their sum, denoted as S + T to be the set of all points x of the form x = y + z, y in S, z in T. The negative of a set S, denoted as –S, consists of all points x of the form x = –y, y in S.

Given two linear spaces Z and U over the same field, their direct sum is a linear space denoted as Z ⊕ U, consisting of ordered pairs (z,u), z in Z, u in U. Addition and multiplication by scalars is componentwise.

Definition. A subset Y of a linear space X is called a linear subspace of X if sums and scalar multiples of Y belong to Y.

Theorem 1.

Exercise 1. Prove theorem 1.

Let S be some subset of the linear space X. Consider the collection {Yσ } of all linear subspaces that contain the setS. This collection is not empty, since it certainly contains X.

Definition. The intersection ∩Yσ of all linear subspaces Yσ containing the set S is called the linear span of the set S.

Theorem2.

(12)

Proof Part (i) is merely a rephrasing of the definition of linear span. To prove part (ii), we remark that on the one hand, the elements of the form (12) form a linear subspace of X; on the other hand, every x of form (12) is contained in any subspace Y containing S.

REMARK 1. An element x of form (12) is called a linear combination of the points x1, … ,xn. So theorem 1 can be restated as follows:

The linear span of a subset S of a linear space consists of all linear combinations of elements of S.

Definition. X a linear space, Y a linear subspace of X. Two points x1 and x2 of X are called equivalent modulo Y, denoted as x1 = x2 (mod Y), if x1 − x2 belongs to Y.

It follows from the properties of addition that equivalence mod Y is an equivalence relation, meaning that it is symmetric, reflexive, and transitive. That being the case, we can divide X into distinct equivalence classes mod Y. We denote the set of equivalence classes as X / Y. The set X / Y has a natural linear structure; the sum of two equivalence classes is defined by choosing arbitrary points in each equivalence class, adding them and forming the equivalence class of the sum. It is easy to check that the last equivalence class is independent of the representatives we picked; put differently, if x1 ≡ z1, x2 ≡ z2. then x1 + x2 ≡ z1 + z2 mod Y. Similarly we define multiplication by a scalar by picking arbitrary elements in the equivalence class. The resulting operation does not depend on the choice, since, if x1 ≡ z1, then kx1 ≡ kz1 mod Y.The quotient set X / Y endowed with this natural linear structure is called the quotient space of X mod Y. We define codim Y = dim X / Y.

Exercise 2. Verify the assertions made above.

As with all algebraic structures, so with linear structures we have the concept of isomorphism.

Definition. Two linear spaces X and Z over the same field are isomorphic if there is a one-to-one correspondence T carrying one into the other that maps sums into sums, scalar multiples into scalar multiples; that is,

(13)

We define similarly homomorphism, called in this context a linear map.

Definition. X and U are linear spaces over the same field. A mapping M : X → U is called linear if it carries sums into sums, and scalar multiples into scalar multiples; that is, if for all x, y in X and all k in F

(14)

X is called the domain of M, U its target.

REMARK 2. An isomorphism of linear spaces is a linear map that is one-to-one and onto.

Theorem 3.

Exercise 3. Prove theorem 3.

A very important concept in a linear space over the reals is convexity:

Definition. X is a linear space over the reals; a subset K of X is called convex if, whenever x and y belong to K, the whole segment with endpoints x, y, meaning all points of the form

(15)

also belong to K.

Examples of convex sets in the plane are the circular disk, triangle, and semicircular disk. The following property of convex sets is an immediate consequence of the definition:

Theorem 4. Let K be a convex subset of a linear space X over the reals. Suppose that x1, … , xn belong to K; then so does every x of the form

(16)

Exercise 4. Prove theorem 4.

An x of form (16) is called a convex combination of x1 ,x2, …...