Analysis On Manifolds

James R. Munkres

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9780429973772
• Edition: 1
• Topic: Mathématiques
• Subtopic: Mathématiques générales

A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.

1

The Algebra and Topology of ℝⁿ

§1. REVIEW OF LINEAR ALGEBRA

Vector spaces

Suppose one is given a set V of objects, called vectors. And suppose there is given an operation called vector addition, such that the sum of the vectors x and y is a vector denoted x + y. Finally, suppose there is given an operation called scalar multiplication, such that the product of the scalar (i.e., real number) c and the vector x is a vector denoted cx.

The set V, together with these two operations, is called a vector space (or linear space) if the following properties hold for all vectors x, y, z and all scalars c, d:

(1) x + y = y + x.

(2) x + (y + z) = (x + y) + z.

(3) There is a unique vector 0 such that x + 0 = x for all x.

(4) x + (–1)x = 0.

(5) 1x = x.

(6) c(dx) = (cd)x.

(7) (c + d)x = cx + dx.

(8) c(x + y) = cx + cy.

One example of a vector space is the set ℝⁿ of all n-tuples of real numbers, with component-wise addition and multiplication by scalars. That is, if x = (x₁,…,x_n) and y = (y₁,…,y_n), then

$\begin{array}{l}\mathbf{\text{x}}\text{+}\mathbf{\text{y}}\text{=}\text{(}{x}_1\text{+}{y}_1\text{,}\dots \text{,}{x}_n\text{+}{y}_n\text{),}\\ c\mathbf{\text{x}}=(c{x}_1,\dots ,c{x}_n).\end{array}$

The vector space properties are easy to check.

If V is a vector space, then a subset W of V is called a linear subspace (or simply, a subspace) of V if for every pair x, y of elements of W and every scalar c, the vectors x + y and cx belong to W. In this case, W itself satisfies properties (1)–(8) if we use the operations that W inherits from V, so that W is a vector space in its own right.

In the first part of this book, ℝⁿ and its subspaces are the only vector spaces with which we shall be concerned. In later chapters we shall deal with more general vector spaces.

Let V be a vector space. A set a₁,…, a_m of vectors in V is said to span V if to each x in V, there corresponds at least one m-tuple of scalars c₁,…, c_m such that

$\mathbf{\text{x}}=c_1{\mathbf{\text{a}}}_1+\cdots +c_m{\mathbf{\text{a}}}_m.$

In this case, we say that x can be written as a linear combination of the vectors a₁,…, a_m.

The set a₁,…,a_m of vectors is said to be independent if to each x in V there corresponds at most one m-tuple of scalars c₁, …, c_m such that

$\mathbf{\text{x}}=c_1{\mathbf{\text{a}}}_1+\cdots +c_m{\mathbf{\text{a}}}_m.$

Equivalently, {a₁,…,a_m} is independent if to the ze...