Functional analysis is that part of mathematical analysis, in which analysis is made, of some linear spaces with their linear operators on a vector space.
In other words, functional analysis is the study of metric spaces, with some basic concepts like neighborhoods, open and closed balls, open and closed sets, limit points, sequences, continuous functions, isometry and many other topics.

## Introductory Functional Analysis with Applications

We first define a metric space as follows:
Let X be a non empty set with X * X = {(x, y) : x belongs to X, and y belongs to X}. Then a real valued function d defined on X * X, i.e.,
d : X * X $\rightarrow$ R is called a metric on X, if it satisfy the following axioms:
M1: d (x, y) >= 0, for all x, y belonging to X.
M2: d (x, y) = 0 if and only if x = y.
M3: d (x, y) = d (y, x), for all x, y belonging to X, this is also known as symmetric property.
M4: d (x, z) <= d (x, y) + d (y , z), for all x, y, z belonging to X, this is also known as triangle inequality.
Then the pair (X, d) is called a metric space.

Note that, the real number d (x, y) is called as the distance between x and y. This means M4 states that the distance d (x, z) of one side of the triangle is less than or equal to the sum of the distances of the other two sides of a triangle as shown below:

This study of metric spaces is then further extended with the concept of complete metric spaces, in which every Cauchy sequence is convergent. And therefore, the study of functional analysis is also related with the real analysis too.

Functional analysis also includes the concept of compactness and connectedness of metric spaces. Hence, functional analysis has many applications with the introduction of Cantor’s intersection theorem, the principle of contraction mapping, the construction of real numbers as the completion of the incomplete metric space of rationals. Functional analysis is also applied by considering Real numbers as a complete ordered field.

Functional analysis can be again extended with the analysis of functions of two and three variables, in which Schwarz’s and Young’s theorems are widely applied. The concept of implicit function theorem with maxima and minima is applied to find out the solutions of optimization problems.
Lagrange’s method of undetermined multipliers is also applied to find out the value of optimized functions.Hence, functional analysis has an important aspect in the field of science and mathematics too. It requires the knowledge of the concept of linear algebra and calculus too, as its base, because the whole functional analysis depends on these topics only.

Some of the important areas of applications of functional analysis are:
1).    Problems regarding the solutions of partial and ordinary differential equations,
2).    In the field of numerical analysis,
3).    Finding the solutions of the problems in calculus of variations,
4).    In the field of approximation theory,
5).    In the field of optimization theory and problems,
6).    To find the equations of integral.

## Elements of Functional Analysis

The concept of functional analysis includes the following elements:
Open and Closed Balls (spheres): Let (X, d) be a metric space and a belongs to X. If r > 0, then the set, {x belongs to X: d (a, x) < r} is called an open ball or open sphere of radius r centered at a. It is denoted as B (a, r) or Sr (a). Thus,
B (a, r) = {x belongs to X: d (a, x) < r}.
Similarly, If r > 0, then the set, {x belongs to X: d (a, x) $\leq$ r} is called a closed ball or closed sphere of radius r centered at a. It is denoted as B * (a, r) or Sr [a]. Thus,
B * (a, r) = {x belongs to X: d (a, x) $\leq$ r}.

Neighborhood of a point: Let (X, d) be a metric space and S is contained in X. The set S is called a neighborhood of a point a belonging to X, if there exists some epsilon e > 0, such that B(a, e) is contained in S.

Open set: A set S is an open set if for each x belonging to S, there exists some e > 0, such that B(x, e) is contained in S.

Interior of a set: A point a belonging to A, is called an interior points of a set A, if there exists some epsilon e > 0 such that B (a, e) is contained in A.
Products: Let (X, d1), (Y, d2) be two metric spaces and let Z = X * Y. Then Z with the metric D defined as
D ((x1, y1), (x2, y2)) = max {d1 (x1, x2), d (y1, y2)} for all (x1, y1), (x2, y2) belonging to Z is called the product of (X, d1) and (Y, d2).
A separable metric space is a space which has a countable dense subset.
The diameter of a set A is defined as d(A) = sup {d (a1, a2): a1, a2 belongs to A}.
A bounded set is a set whose diameter is finite.
The distance between two non empty subsets A and B of a metric space is d(A, B) = inf {d (a, b): a belonging to A, and b belonging to B}.

A sequence < xn> in a metric space (X, d) is said to converge to a point x belonging to X if for each epsilon e > 0, there exists a positive integer m: d(xn, x) < e for all n $\geq$ m.

The L1 space: The set S of all infinite sequences x = < x1, x2, ….> of real number such that sigma mod of xi converges with the metric d defined as:
d (x, y) = sigma form i = 1 to infinity, mod of (xi - yi), for all x, y belonging to S.

The L2 space or The Hilbert Space: The set H of all infinite sequences x = < x1, x2, ….> of real number such that sigma i varying from 1 to infinity, of xi$^ 2$ converges with the metric d: H * H $\rightarrow$ R defined as:
d (x, y) = [sigma form i = 1 to infinity, (xi - yi) 2 ] 2 , for all x, y belonging to H.

## Principles of Functional Analysis

The main principle of Functional analysis is to know about the structure and behavior of functions, because it is not necessary that all functions respond to the similar behaviors. Therefore, functional analysis follows some principles because of its elements and definitions, some of which are mentioned below:Principle of Bolzano Weierstrass Property (BWP):
A metric space (X, d) is said to have BWP if every infinite subset of X has a limit point.
Thus, we get the following important theorem in the study of functional analysis, that is:
In a metric space (X, d), the following are equivalent:
a)    X is compact (every open cover of X has a finite sub cover).
b)    X is sequentially compact.
c)    X has the Bolzano Weierstrass Property.

Principle of Schwarz’s Theorem:

If (a, b) be a point of the domain D contained in R^2 of a real valued function f such that
1)    fx exists in a certain nbd. of (a, b)
2)    fxy is continuous at (a, b)
then fyx (a, b) exists and is equal to fxy (a, b).

Principle of Young’s Theorem:
If (a, b) be a point of the domain contained in R2 of a function f such that fx and fy are both differentiable at (a, b), then fxy (a, b) = fyx (a, b).

Principle of Banach’s Contraction Mapping Theorem or Fixed point Theorem:
This theorem has opened up tremendous opportunities of research in Fixed Point Theory.
The statement of this theorem is as follows:
Every contraction mapping on a complete metric space has a unique fixed point.

Principle of Normed and Banach spaces:
Let V be any vector space and let N be a norm on V. Then the pair (V, N) is called a normed space, which can be related with a distance function, which is then converted into a metric space, and is denoted by (V, !! . !!)
Thus, a normed space is also a normed complex vector space because it satisfies the triangle inequality and the positivity of each element. Since, it satisfies the triangle inequality, hence, the function d(x, y) = mod of [x - y] = mod of [y - x] will be a metric space and when V is complete with respect to the above metric, then V is called as a Banach space.

Principle of Hahn-Banach Theorem:

If X is a normed vector space having R or C as scalars and if Y be any sub space of it . Let ρ is a linear functional which is bounded on Y, then there exists an increased M of ρ to X such that (mod of M) = (mod of ρ).

Principle of Open Mapping Theorem:

Let T : X $\rightarrow$ Y be a continuous linear mapping (surjective) of Banach spaces. Then there will exist φ > 0 in such a way that for all y belonging to Y and mod of y < φ, there is some x belonging to X with mod of x $\leq$ 1 such that Tx = y.