MTH641 Midterm Notes

Metric Spaces: In functional analysis we shall study more general “spaces” and “functions” defined on them. A metric space is a pair (X,d),where X is a set and d is a metric on X (or distance function on X), that is, a function defined on X x y z we have following four properties. 1 Property:

Property:

3rd Property: ( , ) (Sy st M

nd M d x y if a d is real valued finite and non negative

nd only if x y

M d x y d y x mmetry)

4th Property: ( , ) (Triangle Inequality) M d x y d x z d z y The above four properties called axioms of metric space. As metric space is ordered pair so we take mean two elements from set X. Equality (3 points on same line): d x y d x z d z y Inequality (3 points of triangle): d x y d x z d z y

Now if we have more than three points say ,............... n x x x x then distance between any two point say

and x is n n n d x x d x x d x x d x x d x is the generalized triangle inequality.

Subspace:

A subspace (Y,d) of (X,d) is obtained if we take a subset Y and restrict d to Y Y Thus the metric on Y is the restriction Y Y d d d is called the metric induced on Y by d.

Example of Metric Space:

•  Real line ℝ
•  Euclidean plane ℝ2
•  3-Dimensional Euclidean Spaceℝ3
•  Complex Plane ℂ
•  Sequence Space ∞  Functional Space C[a,b]  Space 𝑝  Hilbert Sequence 2

Examples of Metric Space:

Real line ℝ forms a metric Space

Let x and y be two real points on real line, then d x y y x y

Now we prove all the four properties (axioms) of metric space. d x y y

d x y z z y z

d x y z z y

d x z d z y

Euclidean plane ℝ𝟐 forms a metric Space

Euclidean space mean that the points are taken from  in ordered pair.

x y x y

Suppose that one point is x y and the other point is y

thenthe distance d between these two points is

d y y

Thus d is a metric space

Sequence Space l

If we take all sequences of complex number which are bounded in a set then the set is called l  . The

distance function is ( , ) sup i i i d x y

This sequence of complex numbers space, d(x,y) form a metric space.

Function Space C[a, b]

Here C[a,b] is a function space and x and y are functions on t variable and also real valued ( its value is always real value) as x x t and y y t (Real values and continuous).

d(x , y) =0 if x and y are same and d(x , y) =1 if x and y are different. Here the distance function is changed from previous one, the metric d defined by

j j

j j j d x y j

where j j and y and are all complex nos.

i i i i i i i i

i i i i i i i i

i i i i i i

i i i i i i

i i i i i i

i i i i i i i i i

Space

pl

p p j j j p j j j d x y

where y and

The real Space

pl If the elements j j and  are not complex numbers but from real numbers then the

space is called real space pl .

The complex Space

pl If the elements j j and  are complex numbers then the space is called complex

space pl

The Hilbert Sequence Space

2l Now the case p=2 (fixed) The Hilbert sequence space 2l with the metric defined by

j j j

j j j d x y

Open Set and Closed Set on Real Line 

On real line we have open set not open ball. (2,5) is an open set. It includes all values between 2 and 5 but does not include 2 and 5. [2,5] is a closed set. It includes all values between 2 and 5 including 2 and 5.

Open Set and Closed Set on Real Line

In  we have open ball. Here open ball has center ox and radius r. It includes all values but does not

include boundaries

Points on the Boundary:

; on o d x x r x is boundary

All the points are lies on the boundary. If the difference of that point x from the center ox is r. ( o d x r

x is a boundary point.

Points inside the Boundary

All the points are lies inside the boundary if thedistance between x and the center ox (i-e. difference of that

point x from the center ox ) is less than r. o d x r  x lies inside the boundary.

Points outside the Boundary

All the points are lies outside the boundary if the distance between x and the center ox (i-e. the difference

of that point x from the center ox ) is greater than r. ( o d x r x lies outside the boundary.

Open Ball and Closed Ball in

Open Ball:

In  if boundary is not included then it is open ball. It means all points inside the boundary are included. o d x r

In  if boundary is included then it is closed ball. It means all points inside the boundary and on the boundary are included. o d x r

Open Sphere, Closed Sphere in

In  we have open sphere, closed sphere.

Ball and Sphere (General Form)

Open Ball:

o o B x r X d x x r

o o B x r X d x x r

Sphere:

o o S x r X d x x r

Sphere: Sphere includes all those points which are exactly lies on the boundary or on the radius r. It does not have any inside or outside points. In discrete space, we have defined distance function. Sphere can be empty.

( , y) 1 ; x

r

r c

c d x x

d x y

d x x

d x x Sphere =

For boundary we can subtract open ball from closed ball. S x r B x r B x r

Open Set:

A subset M of a metric space X is said to be open if it contains a ball about each of its points.

Closed Set:

A subset K of X is said to be closed if its complement (in X) is open. that is c K K is open.

Neighborhood of a Point: We can find an open ball around each point in Open set. “An open ball , ) of radius o B x   is often called an -neighborhood of ox .” By a neighborhood of ox ,

we mean any subset of X which contains an -neighborhood of ox Difference between Radius r and For radius r means larger values, 0.1, 0.5, 10,40 while radius  means

very small values like 0.002, 0.000003 etc. Interior Point: We call ox an interior point of a set M if M is a neighborhood of ox . The interior of M

is the set of all interior points of M. Int(M) is open and is the largest open set contained in M. Collection of all open balls is an open set whether the radius of open ball is greater or smaller.

Topological Space:

Let  be collection of all open subsets of X. Then (X,  ) is said to be a topological space if it satisfies following properties: T1): and X T2): The union of any member of  is a member of  . T3): The intersection of finitely many members of  is a member of  .

Continuous Mappings: Let X d and Y Y d  be metric spaces.

A mapping T X Y is said to be continuous at a point ox

if for every there is a o such that d Tx Tx for all x satisfying ( , o d x x

Continuous Mapping: A mapping T of a metric space X into a metric space Y is continuous if and only if the inverse image of any open subset of Y is an open subset of X. Conversely we consider inverse image of open subset is open and prove that it is continuous.

Proof:

Suppose a mapping T X Y and T is continuous. Now we will prove that inverse image of any open subset in Y is open in X. Let S Y be open subset. Let oS be the inverse image of S. Space X  spaceY

We have to prove that this oS open. For this we have two cases.

1st case:

Suppose that we have chosen the element has no inverse image then open oS (because empty set is always open)

2nd case:

Now suppose oS is not empty then there is at least one

Point ox such that

o o o S S