Lecture
Background materials
1 The set of real numbers
Calculus is a branch of mathematics dedicated to exploring the characteristics of “functions.” From a
physical perspective, a function refers to the assignment o f a scalar value to each point within a set,
known as the domain or focal set. The values collectively constitute the range of this assignment or
function This assignment can be represented as f: D
f
!R
f
, where D
f
is the domain of the function,
and R
f
is the range. Generally, the domain of a function is a subset D of R
n
, where R
n
represents
the set of all n-tuples f(x
1
; :::; x
n
)g, with each x
j
denoting a point in R.
To grasp the intricacies of functions, the focal point of Calculus, it is essential to initially comprehend
the properties of a function's domain and range. In this context, we introduce the space R
n
and
delve into its algebraic and topological properties.
The set of real numbers, denoted by R, serves as a mathematical representation of one-dimen-
siona l “continuum” objects, conceptualized as “straight” lines. The term “continuum” implies the
theoretical possibility of inﬁnite divisions within the object. Straightness, a geometric property
easily grasped by all, refers to lines devoid of curvature. This perception gives R a geometric
interpretation as a straight line, commonly referred to as the x-axis or y-axis in the xy-plane or in
thre e-dimensional space.
Similarly, R
2
can be envisioned as an unbounded straight plane without curvature, and R
3
rep-
resents space without curvature, extending this pattern further. The elements o f R
n
have a dual
nature; they can be viewed both as points x=(x₁; :::; xₙ) and alternatively as vectors x~ =(x₁; :::; xₙ).
These two interpretations lead to distinct structures on R
n
.
1.1 R
n
as a vector space
As a set of vectors, R
n
b ecomes a vector space through the introduction of two operations: vector
addition and scalar multiplication. For two vectors x~ = (x₁; :::; xₙ) and y~ =(y ₁; :::; yₙ), these opera-
tions are deﬁned as follows:
x~ + y~ = (x
1
+ y
1
; : : : x
n
+ y
n
):
Multiplying by a scalar λ, means
λ y
~
= (λx
1
; : : : ; λx
n
):
2