[] Infinite set of dual linear forms

These notes to demonstrate the implicit role that the functional play in the field of theoretical physics. Starting from its role in the
foundations of the classical dynamics, the relativistic theories of space and time, quantum mechanics and field theories.
Vertually everywhere, the idea of the functional space was there. Functions and geometry needed to be supplemented by transfomations that depend on a continuous set of parameters, that meant that it needed an infinite set of functions and their duals.

[] The variational principles

A functional principle -implies the set of coordinates and translations -variations; the invariance of the functional; the symmetries and the conservations.

[] The canonical transformations

The symmetry of the coordinates and momenta; transformations performed by the functional.

[] The symmetries of the Lagrangian

Dependence of physical quantity on the symmetries of a more abstract geometric one. Energy momentum on time space.

[] The optics mechanics analogy

Associate to every geometric optical quantity a dynamical variable.

[] The Minkowski space time

Transformations required infinity of spaces; the generators of the transformations are their duals. In quantum mechanics the generator is the energy momentum form, that makes these infinity into one functional space.

[] General relativity

Each point of the manifold has its own local inertial frame; the duals connect the neibouring points, and this again is a functional space.

[] Gauge theory

Phase is the geometric dual to action; the continuous set of the parameters generates a functional space.

[] Functions, geometry and hydrodynamic models

The continuous transformations; duals: are needed to supplement functions and geometry.

[] Quantum mechanics

Action reemphasized; the abstract functional structure; transformtions, the relationship of particles, waves and propability; of space time and energy momentum;

[] Transformation theory

The symmetry; the action.

[] Renormalization theory

The continuous set of the parameters; the role of the functional.

[] Information theory

Functional spaces have exponential information capacity. This is connected with quantum mechanics.

[] Statistical physics

Deriving the distributions from a single mathematical entity. The entity is the functional.

[] The S matrix

The infinite set; is the basic idea behind the folowing terms: Feynman Integral, Universal Wave Function, String Theory.

[] Feynman integral

The functional is expressed more explicitly but missinterpreted.

[] The universal wave function

Function space; nondual!. This is the mean weakness of the approach.

[] String theory

Infinite set of points; infinite set of oscillations; too classical a model. The basis is the S matrix.