# Theory of Function Spaces: Course Outline (MAT-621)

A function space is a topological space whose points are functions. There are many different kinds of function spaces, and there are usually several different topologies that can be placed on a given set of functions.

## Course Outline

Vector and norm spaces​

Lebesgue sequence and function spaces: Lebesgue sequence spaces and related theorems. Hardy and Hilbert inequalities. Essentialy bounded functions, Lebesgue spaces (p>1), Embeddings, Approximation in Lebesgue spaces, Duality, Reflexivity, weak convergence and continuity of translation operator, weighted Lebesgue spaces, Isometries, Lebesgue spaces (0<p<1).

Distribution Function and Rearangements: Distribution functions, Decreasing Rearangements, Rearanagement of Fourier Transform.

Weak Lebesgue spaces: Weak Lebesgue spaces, Convergence in Measure, Interpolation and Normability.

Lorentz Spaces: Lorentz space and Normability, Completeness, Separbility, Duality, Lorentz sequence spaces.

Non-Standard Function spaces: Variable Lebesgue space Definition and their properties. Some differences between constant exponent and variable exponent spaces.  Grand Lebesgue space, Amalgam and Herz type spaces with variable exponents.