Real space to Metric space and Normed linear space
Real space to Metric space and Normed linear space
M. Velrajan
Any Pure Mathematics learners after learning the Real space learn the Euclidean Space, Metric Space and Normed Linear Space. In this we discussed how to understand Euclidean Space, Metric Space and Normed Linear Space as generalisations of Real Space.
First we recall
1. The (archimedean) absolute value of ℝ is the function ℝ → ℝ+ defined by
|x|= { −x, if x < 0; x, if x ≥ 0 .
For x ∈ ℝ, |x| is called the absolute value or modulus of x.
2. Geometrically, for x ∈ ℝ, |x| is the distance of x from the origin on the real line.
3. For x ∈ ℝ and r > 0,
|x| < r if and only if – r < x < r
4. For x ∈ ℝ, |x| = (x2)1/2, the non-negative square root of x2
5. If x, y ∈ ℝ then
|x| ≥ 0 and |x| = 0 if and only if x = 0
|x| = |- x|
|x + y| ≤ |x| + |y|
|x y| = |x| |y|
6. If x, y ∈ ℝ then |x - y| = ((x - y)2)1/2 is the distance between
x and y on the real line.
7. If x, y, z ∈ ℝ then
|x - y| ≥ 0 and |x - y| = 0 if and only if x = y
|x - y| = |y - x|
|x - y| ≤ |x - z| + |z - y|
8. If x, a ∈ ℝ and r > 0 then
|x - a| < r if and only if – r < (x - a) < r
if and only if a – r < x < a + r
if and only if x ∈ (a - r, a + r) = B(a, r) = {y / |y - a| < r}
9. If x, y ∈ ℝ then | |x| - |y| | ≤ |x - y|
For,
|x| = |x - y + y| ≤ |x - y| + |y|
|x| - |y| ≤ |x - y|
|y| - |x| ≤ |y - x| = |x - y|
| |x| - |y| | ≤ |x - y|
10. The function |x| is a continuous function.
For,
For ϵ > 0, if |x - y| < ϵ then | |x| - |y| | ≤ |x - y| < ϵ
Hence the function |x| is a continuous function.
11. A sequence (xn) of real numbers is said to converge to the real number x
if for each ϵ > 0 there exists a natural number N such that |xn − x| < ϵ for all n ≥ N
i.e. x - ϵ < xn < x + ϵ or xn ∈ (x - ϵ, x + ϵ) = B(x, ϵ) = {y / |y - x| < ϵ}, for all n ≥ N.
If (xn) converges to x, we write limn→∞ xn = x, or xn → x.
The number x is called the limit of the sequence (xn).
12. Observe that sequence (xn) of real numbers converges to x
if and only if for each ϵ > 0 there exists a natural number N
such that |xn − x| < ϵ for all n ≥ N
if and only if (|xn − x|) converges to 0
13. If sequences of real numbers (xn) and (yn) converge to x and y respectively then (xn + yn) converges to x + y,
(xn - yn) converges to x - y and
(xnyn) converges to xy
i.e. the addition and the multiplication of ℝ are continuous functions.
14. For x = (x1, x2, . . . , xn) ∈ ℝn, ||x|| = (x12 + x22 + . . . + xn2)1/2 ,
the non-negative square root of x12 + x22 + . . . + xn2.
See how ||x|| is defined for x = (x1, x2, . . . , xn) ∈ ℝn, by extending |x| for x ∈ ℝ.
Note that ℝ is a field and ℝn, n > 1, is a commutative ring with identity, under coordinate wise addition and multiplication, but it is not a field.
Also ℝn is a vector space over ℝ and ℝn is called Euclidean space.
15. Geometrically, for x = (x1, x2, . . . , xn) ∈ ℝn, n ≤ 3,
||x|| is the distance of x from the origin.
16. For x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn) ∈ ℝn,
||x|| ≥ 0
||x|| = 0 if and only if x = 0
||x|| = ||- x||
||x + y|| ≤ ||x| + ||y||
17. For x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn) ∈ ℝn,
||x - y|| = [(x1 - y1)2 + . . . + (xn - yn)2]1/2 is the distance between x and y
18. For x, y, z ∈ ℝn,
||x - y|| ≥ 0
||x - y|| = 0 if and only if x = y
||x - y|| = ||y - x||
||x - y|| ≤ ||x - z|| + ||z - y||.
Geometrically in ℝ2, the last inequality says that the length of the side joining x and y is less than or equal to the sum of lengths of the other two sides of the triangle with x, y, z as the vertices.
Hence the inequality or any general similar inequality is called the triangle inequality.
19. Compare the properties of ||x||, x ∈ ℝn with |x|, x ∈ ℝ.
Both have the same properties.
Whatever defined and proved in the real space ℝ using only the properties of
|x| or |x - y|, can be defined and proved in the Euclidean space ℝn by replacing |x| or |x - y| with ||x|| or ||x - y||.
For example, a function f : ℝm → ℝn is continuous at x ∈ ℝm
if for any ϵ > 0 there exists δ > 0 such that
if ||x - y|| < δ then ||f(x) - f(y)|| < ϵ.
Note that ||x - y|| is the norm in ℝm and ||f(x) - f(y)|| is the norm in ℝn.
20. For (x, y), (x′, y′) ∈ ℝ × ℝ, | |x - y| - |x′ - y′| | ≤ |x - x′| + |y - y′|
and the function (x, y) → |x - y| is a continuous function of ℝ × ℝ into ℝ+.
For,
|x - y| ≤ |x - x′| + |x′ - y|
≤ |x - x′| + |x′ - y′| + |y′ - y|
|x - y| - |x′ - y′| ≤ |x - x′| + |y - y′| and
|x′ - y′| - |x - y| ≤ |x′ - x| + |y′ - y| = |x - x′| + |y - y′|
Hence
| |x - y| - |x′ - y′| | ≤ |x - x′| + |y - y′|
≤ 21/2[(x - x′)2 + (y - y′)2]1/2
= 21/2||(x, y) - (x′, y′)||
[If 0 ≤ a + b then (a + b) ≤ 21/2(a2 + b2)1/2
For,
(a + b)2 - 2(a2 + b2) = - (a - b)2 ≤ 0.
(a + b)2 ≤ 2(a2 + b2)
Taking the nonnegative square root on both sides,
(a + b) ≤ 21/2(a2 + b2)1/2
Find where, 0 ≤ a + b is necessary]
For ϵ > 0, if ||(x, y) - (x′, y′)|| < ϵ/21/2 then | |x - y| - |x′ - y′| | < ϵ.
The function (x, y) → |x - y| is a continuous function of ℝ × ℝ into ℝ+
21. Let X be a nonempty set.
A function d from X × X into ℝ+ is called a metric if
d(x, y) ≥ 0 and
d(x, y) = 0 if and only if x = y
d(x, y) = d(y, x)
d(x, y) ≤ d(x, z) + d(z, y).
The set X together with the function d is called a metric space.
22. When we learn a metric space after real space ℝ, we should have compared the properties of |x - y|, the distance between the real numbers x and y, given in 7, with the definition of a metric.
|x - y| is replaced with d(x, y).
So the following have to be observed :
a) A metric is a generalisation of |x - y|, the distance between the real numbers x and y, to any nonempty set X.
b) ℝ together with the distance between the real numbers is a metric space.
c) Whatever defined and proved in the real space ℝ using only the properties of |x - y| stated in 7, can be defined and proved in a metric space by replacing |x - y| with d(x, y)
d) Whatever defined and proved in a metric space can be defined and proved in the real space ℝ, but not conversely.
Following are some examples :
One can have more examples.
23. we can define convergence of a sequence in a metric space as :
A sequence (xn) in a metric space X is said to converge to x ∈ X
if for each ϵ > 0 there exists a natural number N such that d(xn, x) < ϵ for all n ≥ N
i.e. xn ∈ B(x, ϵ) = {y / d(y, x) < ϵ}, for all n ≥ N.
But not x - ϵ < xn < x + ϵ, for all n ≥ N, since in general,
there is no order relation < in a metric space and real number ϵ cannot be added or subtracted with an element x of the metric space X.
24. We can observe, by replacing |x - y| with d(x, y) in 12, that :
In a metric space (X, d), (xn) converges to x if and only if (d(xn, x)) converges to 0 in the real space.
25. We can prove, by replacing |x - y| with d(x, y) in 20, that,
If (X, d) is a metric space then for (x, y), (x′, y′) ∈ X × X,
| d(x, y) - d(x′, y′) | ≤ d(x, x′) + d(y, y′)
≤ 21/2 [(d(x, x′))2 + (d(y, y′))2]1/2
and the metric function
(x, y) → d(x, y) is a continuous function of X × X into ℝ+.
26. Since in general, there is no binary operation + or multiplication in a metric space, we cannot add, multiply two sequences in a metric space.
Recall
27. A vector space N over the real or the complex field is said to be a normed linear space (nls) if for each x ∈ N, there exists a nonnegative real number ||x||, called norm x, such that
||x|| ≥ 0
||x|| = 0 if and only if x = 0
||𝛂x|| = |𝛂| ||x||, for all scalar 𝛂
||x + y|| ≤ ||x| + ||y||.
28. When we learn a normed linear space after real space ℝ and metric space,
we should have compared the properties of |x| given in 5, with the properties of ||x|| and recall that how the properties of |x| are used in |x - y|, the distance between the real numbers x and y, so that ℝ is a metric space.
|x| is replaced with ||x||.
So the following have to be observed :
a) A norm is a generalisation of |x|, the absolute value of real number x, to any vector space N over the real or the complex field.
b) ℝ together with |x|, the absolute value of real number x, is a normed linear space. In fact ℝn together with ||x|| is a normed linear space.
c) Whatever defined and proved in the real space ℝ using the properties of |x| stated in 5 and the properties of vector space ℝ, can be defined and proved in a normed linear space by replacing |x| with ||x||, in particular,
a normed linear space is a metric space with the metric d(x, y) = ||x - y||.
d) Whatever defined and proved in a normed linear space can be defined and proved in the real space ℝ, but not conversely.
e) Whatever defined and proved in a metric space can be defined and proved in a normed linear space by replacing d(x, y) with ||x - y||
Following are some examples :
29. We can define convergence of a sequence in a normed linear space as :
A sequence (xn) in a normed linear space X is said to converge to x ∈ X
if for each ϵ > 0 there exists a natural number N such that ||xn - x|| < ϵ for all n ≥ N
i.e. xn ∈ B(x, ϵ) = {y / ||y - x|| < ϵ}, for all n ≥ N.
But not x - ϵ < xn < x + ϵ, for all n ≥ N, since in general, there is no order relation < in a normed linear space and real number ϵ cannot be added or subtracted with an element x of the normed linear space X.
30. We can observe, by replacing |x - y| with ||x - y|| in 12, that :
In a normed linear space N,
(xn) converges to x if and only if (||xn - x||) converges to 0 in the real space.
31. We can prove, by replacing |x| with ||x|| in 9 and 10, that,
| ||x|| - |y|| | ≤ ||x - y|| and the function ||x|| is a continuous function.
32. In a normed linear space N, if (xn) converges to x and (yn) converges to y and (𝛂n) converges to 𝛂 in the scalar field then
(xn + yn) converges to x + y,
(xn - yn) converges to x - y and
(𝛂nxn) converges to 𝛂x
i.e. the addition and the scalar multiplication of a normed linear space are continuous functions.
If we learn Pure Mathematics in a similar way we can understand how the concepts are extended or generalised. And also we can think of such extension or generalisation of what we learned.
Learn Pure Mathematics in an enjoyable way.
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