Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach. In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theories of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis. The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product.

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Related exercises (32)

Changing the vector frame and the coordinate system

Let v⃗ be a vector whose coordinates in the vector frame Oxyz are $\left( \begin{array}{c} v_1\\ v_2\\ v_3\\ \end{array}\right)$ . How would you srite this vector in the alternate vector frame Ox′y′z′ in which the z and z′ axes are identical and in which θ is the angle between x and x′ and between y and y′?
What are the radius vector $\overrightarrow{OP}$ ’s projections on the Cartesian axes in the figures below, as a function of the cylindrical (ρ, ϕ, z) or spherical (r, θ, ϕ

Car in a bend

A sports car is driving around a circular track of radius R with a horizontal velocity of constant norm v. The track is on an incline with angle α from the the horizontal, such that the car is tilted inwards. The static friction coefficient between the tires and the track is μ.

Draw a diagram, considering the vertical plane perpendicular to the car’s velocity and represent all forces acting on the car.
What’s the condition on v for the car not to drift outwards ?

Numerical application: $g=10\

Hole-in-the-table

A horizontal table has a hole that allows a thread of length L to pass without friction. This thread connects two blocks: 1) a block of mass m that slides without friction on the table and whose initial velocity is not in the thread’s direction and 2) a block of mass M that’s vertically hanging below the table. The thread is always taut.

Write the blocks’ total mechanical energy, considering that the thread has constant length.
What are the conserved quantities, that is, the primary integrals

Point on a cylinder

A material point moves on an infinitely long cylinder of radius R. The material point is attracted to a point O on the cylinder’s axes by a force that’s proportional to the distance between the material point and O. There is neither gravity nor friction.

List the forces acting on the material point. Draw these forces on a diagram, as well as the frame of reference associated with the chosen coordinate system.
Write the equations of motion and project them on the frame of reference.

The ladder

A ladder with mass m and length L is resting on a vertical wall. The angle between the wall and the ladder is α and the friction coefficient between the ladder’s foot and the floor isxa0μ. There is no friction between the wall and the ladder. What’s the maximal value

\alpha_{\rm max}

angle α can take such that the ladder’s bottom doesn’t slip on the floor and fall? We consider the ladder as the system, seen from the referential of the floor and the wall. We choose a Cartesian reference frame with a horiz

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