Banach Space

What is Functional Analysis?

Functional Analysis plays an important role in both applied science as well as in mathematics itself. As a result, it is frequently recommended to introduce the student to the area early in the study. It became a distinct area in the 20th century when it was discovered that many mathematical operations, from arithmetic to calculus procedures, share a number of highly similar characteristics.

Here we mainly cover the definition of Functional Analysis and Examples. Also, we discuss different branches of Functional Analysis like  Normed spaces, Banach Space, Inner product space, Euclidean space, Uniform boundness theorem, Spectral theorem, Hahn–Banach theorem, Open mapping theorem, and Closed Graph Theorem.

Definitions:

Definitions; 1) “The branch of mathematics that deals with the theory of vector space and linear algebra” 2) “Functional analysis is a strategy for methodically examining links between problematic behavior and external events” 3) “Functional analysis is a technique for explaining how a complex system works”.

Originally derived from classical analysis, functional analysis is an abstract area of mathematics.  Applications involving ordinary and partial differential equations, numerical analysis, calculus of variations, approximation theory, integral equations, and others provided the impetus for the development of the field.

Metric Space

Definition: A metric space is a pair of (X, d) where X is the set and ” d ” is metric on set X ( or distance function on X ), that is function defined on X x X such that for all x, y, z ∈ X.

Metric Space Axioms

• d is a real-valued, finite and non-negative
• 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)

Examples:

• Real line R. This is the set of real numbers, taken with the usual metric defined by

d(x, y) = | x – y |

• Euclidean Plane R². The metric space R² called the Euclidean plane is obtained if we take the set of ordered pairs of real numbers, written x = ( ξ₁, ξ₂ ) and y = ( η₁, η₂ ), etc.

Normed Space

Definition: A normed space X is a vector space with a norm defined on it.

A normed on a ( real or complex ) vector space X is a real-valued function on X whose value at an x ∈ X is denoted by

|| x ||

Normed Space Axioms

•  || x || ≥ 0
• || x || = 0 ⇔ x = 0
• || αx || = | α | || x ||
• || x + y || ≤ || x || || y ||    ( Triangular inequality )

Here x and y are arbitrary vectors in X and α is any scalar.

A norm on X defines a metric d on X which is given by

d (x, y ) = || x – y ||

and is called the metric induced by the norm. The normed just defined is denoted by ( X, ||.|| ) or simply by X.

Examples:

• n-dimensional Euclidean space is a normed space.

Banach Space

Definition: A Banach space is a completely normed space ( X, ||.|| )

If the norm-induced metric ” d ” is a complete metric, then the normed space (X, ||.||) is a Banach space. If (X, ||.||) is a Banach space, the nome ||.|| of a normed space (X, ||.||) is referred to as the complete norm.

Examples

• Space l^∞ is a Banach space with the norm given by

||x|| = sup _j | ξ_j |

• Space C[a, b] is a Banach space with the norm given by

||x|| = max _{t ∈ j} | x(t) |,

where j = [a, b].

Inner Product Space

Definition: An inner product space ( or pre-Hilbert space ) is a vector space X with an inner product defined on X.

Inner Product Space Axioms:

• 〈 x + y, z 〉 = 〈 x, z 〉 + 〈 y, z 〉.
• 〈 αx, y 〉 = α 〈 x, y 〉.
•    〈 x, x 〉 ≥ 0
• 〈 x, x 〉 = 0 ⇔ x = 0

An inner product on X defines a norm on X given by

|| x || = √〈x, x 〉

Hilbert Space

Definition: A Hilbert space is a complete inner product space.

The addition of a certain distance function provides the final structure that completes it.

d(x, y) = || x – y || = √〈x -y, x – y〉

Point To Be Noted:

• Inner product spaces are normed spaces, and Hilbert spaces are Banach spaces

Examples:

• Euclidean Space Rⁿ. The space Rⁿ is a Hilbert space with the inner product defined by

〈x, y 〉 = ξ₁ η₁ + ξ₂ η₂ + … + ξn η_n

Where x = ( ξ_j ) = ( ξ₁, … ξ_n ) and y = ( η_j ) = ( η₁, …. η_n )

• Unitary Space Cⁿ. The space Cⁿ is a Hilbert space with the inner product given by

〈x, y 〉 = ξ₁ η_{1^– +…  +} ξ_n η_n^–

Uniform Boundness Theorem

Statement: Let ” X ” be Banach space and ” Y ” be a norm linearly space and ” M ” be a subset of B (X, Y) such that

〈Tx: T ∈ M 〉 is a bounded subset of ” Y ” ∀ x ∈ X then ” M ” is a bounded subset of B(X, Y). i.e if ” M ” is pointwise bounded then it is uniformly bounded.

Spectral Theorem

Statement: Let ” T ” be an operator on a finite-dimensional complete Hilbert space ” H “. Let  λ₁, λ₂, … λ_m be distinct eigenvalues of ” T “. Let P₁, P₂, … P_m be corresponding projections on subspaces M₁, M₂, … M_m respectively. The following statements are equivalent

• The M_i‘s are piecewise orthogonal and spans ” H “.
• The P_i‘s are piecewise orthogonal

P₁ + P₂ + … + P_m = I  & T = λ₁ P₁ + λ₂ P₂ + …. + λm P_m

• T is normal

Hahn-Banach Theorem

Statement: Let X be a real vector space and p a sublinear functional on X. Furthermore, let f be a linear function that is defined on a subspace Z of X and satisfies

f(x) ≤ p(x)    ∀ x ∈ Z      → ( A )

Then f has a linear extension of f^– from Z to X satisfying

f^– (x) ≤ p(x)  ∀ x ∈ Z

that is, f^– is a linear functional on X, satisfies ( A ) on X f^– (x) = f(x) for every x ∈ X.

Open Mapping Theorem

Statement: Let X and Y be Banach spaces and let

T: X → Y

be a continuous ( bounded ) linear transformation from X to Y. Then T is open mapping.

i. e For all open subsets A of X, T(A) is an open subset of Y.

Closed Graph Theorem

Statement: Let X and Y be Banach spaces and

T: D( T ) → Y a closed linear operator, Where D ( T ) ⊂ X. Then if D ( T ) is closed in X, the operator T is bounded.

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