The Euclidean Vector Space Rd

March 23, 2017 1 / 14 Points and operations of Rd

Deﬁnition 3.1 ∗ d Let d ∈ N . We denote the Cartesian product R × ... × R by R , and we | {z } d times call its elements points or vectors. If x = (x1,..., xd ), then the number xi is called the ith coordinate or component of the point or vector x.

2 3 Examples: R , R . Deﬁnition 3.2 d We deﬁne addition, multiplication by scalars and scalar product in R as d follows: if v = (v1,..., vd ), w = (w1,..., wd ) ∈ R and λ ∈ R, then

v + w := (v1 + w1,..., vn + wd ), λv := (λv1, . . . , λvd ),

hv, wi := v1w1 + ··· + vnwd .

March 23, 2017 2 / 14 Rd as a Euclidean vector space

Theorem 3.3 d The set R , together with the addition, scalar multiplication and scalar product deﬁned above, is a Euclidean vector space. The vectors

e1 := (1, 0,..., 0), e2 := (0, 1, 0 ..., 0),..., en := (0,..., 0, 1).

d form an orthonormal basis of R , which is called its canonical basis.

March 23, 2017 3 / 14 The Euclidean norm

Deﬁnition 3.4 d By the Euclidean norm or length of a vector v ∈ R we mean the number kvk := phv, vi.

Theorem 3.5 (properties of the norm) d If v, w ∈ R and λ ∈ R, then (i) kvk ≥ 0, and kvk = 0 ⇐⇒ v = 0; (ii) kλvk = |λ|kvk; (iii) kv + wk ≤ kvk + kwk.

March 23, 2017 4 / 14 The Euclidean distance

Deﬁnition 3.6 d By the distance of the points p, q ∈ R we mean the number d(p, q) := kp − qk.

Theorem 3.7 (properties of the distance) d If p, q, r ∈ R , then (i) d(p, q) ≥ 0, ´es d(p, q) = 0 ⇐⇒ p = q; (ii) d(p, q) = d(q, p) (symmetry); (iii) d(p, r) ≤ d(p, q) + d(q, r) (triangle inequality).

March 23, 2017 5 / 14 Sequences in Rd

Deﬁnition 3.8 d A sequence in R is a mapping

d a: N → R

notation: the nth point in the sequence is an := a(n). We refer to the whole sequence as (an)n∈N.

Example

an = (n, 2n).

March 23, 2017 6 / 14 Limits of sequences in Rd

Deﬁnition 3.9 d A sequence (an)n∈N = (an1, an2,..., and ) of points in R is convergent if the sequences of its coordinates

(an1)n∈N, (an2)n∈N,..., (and )n∈N

(as sequences of real numbers) are convergent. Then the limit of (an)n∈N is the vector ( lim an1, lim an2,..., lim and ). n→∞ n→∞ n→∞

Example 1 1 1 n an = ( n , 1 − n , ( 2 ) ).

March 23, 2017 7 / 14 Limits of sequences in Rd

Lemma 3.10 d A sequence (an)n∈N = (an1, an2,..., and ) of points in R converges to a d point A ∈ R if and only if the sequence of the distances

d(an, A)

converges to zero.

March 23, 2017 8 / 14 Multivariable functions and vector-valued functions Definition 3.11 By a (real-valued) function of d variables we mean a function d f : E ⊂ R → R. Example: f (x, y, z) = x2 sin(y + z3) Definition 3.12 By a vector-valued function of one variable we mean a function s f : E → R . where E is a subset of R. Remark: such functions are also called curves. Example: f (t) = (3t, t2, sin(t)). Definition 3.13 By a vector-valued function of d variables we mean a function d s f : E ⊂ R → R . Example: f (x, y) = (x/y, xy, y).

March 23, 2017 9 / 14 Limits of multivariable functions

Accumulation points and limits of functions are deﬁned the same way as in R. Deﬁnition 3.14 d d Let E ⊂ R . A point x ∈ R is an accumulation point of E, if there is a sequence in E \{x} that converges to x.

Deﬁnition 3.15 d Let E ⊂ R and let f : E → R. We say that the limit of f at a point 0 x0 ∈ E is A, if for any sequence (xn)n∈N in E \{x0} converging to x0, we have lim f (xn) = A. n→∞

March 23, 2017 10 / 14 Continuity of multivariable functions

Deﬁnition 3.16 d Let E ⊂ R and let f : E → R. We say that f is continuous at a point 0 x0 ∈ E , if for any sequence (xn)n∈N in E converging to x0, we have

lim f (xn) = f (x0). n→∞

Example The usual operations of R, as two-variable functions f (x, y) = x + y, f (x, y) = xy, x f (x, y) = y , f (x, y) = xy . are continuous on their domain.

March 23, 2017 11 / 14 Continuity of vector-valued functions

d s A vector-valued function f : E ⊂ R → R can be represented with its coordinate functions:   f1(x)  .  f (x) =  .  . fs (x)

Deﬁnition 3.17 A vector-valued function is continuous if its coordinate functions are continuous.

Lemma 3.18 Composition of continuous functions are continuous. So, if d s s t f : E ⊂ R → R and g : F ⊂ R → R are continuous, and f (E) ⊂ (so the composition g ◦ f makes sense) then g ◦ f is continuous.

March 23, 2017 12 / 14 Open sets in Rd

Deﬁnition 3.19 d The open ball of radius r > 0 with center p ∈ R is the set

d Br (p) = {x ∈ R | d(p, x) < r}.

Deﬁnition 3.20 d Let A ⊂ R . Then x ∈ A is an inner point of A if there is an open ball with center x contained in A.

Deﬁnition 3.21 d A ⊂ R is open if all of its points are inner points.

March 23, 2017 13 / 14 Example

– Br (p) is open if r > 0. d – ∅, R are open. 1 – In R = R, the open intervals are open. – A set of a single point is not open. 2 3 – A line in R , R ,... is not open.

Theorem 3.22 The union of open sets is open. The intersection of ﬁnitely many open sets is open.

March 23, 2017 14 / 14