A report on Functional analysis

One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.

Branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense.

- Functional analysis
One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.

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The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.

Hilbert space

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In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.
David Hilbert
The overtones of a vibrating string. These are eigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1, 1⁄2, 1⁄3, ... form the (musical) harmonic series.
The path of a billiard ball in the Bunimovich stadium is described by an ergodic dynamical system.
Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top)
Spherical harmonics, an orthonormal basis for the Hilbert space of square-integrable functions on the sphere, shown graphed along the radial direction
The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy.

The success of Hilbert space methods ushered in a very fruitful era for functional analysis.

Addition of functions: The sum of the sine and the exponential function is

Vector space

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[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

Addition of functions: The sum of the sine and the exponential function is
A typical matrix
Commutative diagram depicting the universal property of the tensor product.
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).

At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.

Stefan Banach

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Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians.

Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians.

Otto Nikodym and Stefan Banach Memorial Bench in Kraków, Poland (sculpted by Stefan Dousa)
Scottish Café, meeting place of many famous Lwów mathematicians
Banach's grave, Lychakiv Cemetery, Lviv (Lwów, in Polish)
Decomposition of a ball into two identical balls - the Banach–Tarski paradox.
Banach monument, Kraków

He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics.

A family of neighborhoods of the origin with the above two properties determines uniquely a topological vector space. The system of neighborhoods of any other point in the vector space is obtained by translation.

Topological vector space

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A family of neighborhoods of the origin with the above two properties determines uniquely a topological vector space. The system of neighborhoods of any other point in the vector space is obtained by translation.

In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.

Hahn–Banach theorem

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The Hahn–Banach theorem is a central tool in functional analysis.

Banach space

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In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.

Function space

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Set of functions between two fixed sets.

Set of functions between two fixed sets.

In functional analysis the same is seen for continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a topology; the best known examples include Hilbert spaces and Banach spaces.

In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Linear algebra

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Branch of mathematics concerning linear equations such as:

Branch of mathematics concerning linear equations such as:

In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

Unitary operator

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In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product.

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

Mathematical analysis

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Branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

Branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.