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 ((x_{1}, y_{1}), (x_{2}, y_{2})) = max {d1 (x_{1}, x_{2}), d (y_{1}, y_{2})} for all (x_{1}, y_{1}), (x_{2}, y_{2}) 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 (a_{1}, a_{2}): a_{1}, a_{2} 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 < x_{n}> 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(x_{n}, x) < e for all n $\geq$ m.

The L1 space: The set S of all infinite sequences x = < x_{1}, x_{2}, ….> of real number such that sigma mod of x_{i} converges with the metric d defined as:

d (x, y) = sigma form i = 1 to infinity, mod of (x_{i} - y_{i}), for all x, y belonging to S.

The L2 space or The Hilbert Space: The set H of all infinite sequences x = < x_{1}, x_{2}, ….> of real number such that sigma i varying from 1 to infinity, of x_{i}$^ 2$ 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.