A fundamental characteristic of many mathematical spaces (e.g. vector spaces, metric spaces, topological spaces, etc.) is their *dimension*, which measures the “complexity” or “degrees of freedom” inherent in the space. 5,110 more words

## Tags » 245C - Real Analysis

#### 245C, Notes 5: Hausdorff dimension (optional)

#### 245C, Notes 4: Sobolev spaces

As discussed in previous notes, a function space norm can be viewed as a means to rigorously quantify various statistics of a function . For instance, the “height” and “width” can be quantified via the norms (and their relatives, such as the Lorentz norms ). 5,689 more words

#### 245C, Notes 3: Distributions

In set theory, a function is defined as an object that *evaluates* every input to exactly one output . However, in various branches of mathematics, it has become convenient to generalise this classical concept of a function to a more abstract one. 6,928 more words

#### 245C, Notes 2: The Fourier transform

In these notes we lay out the basic theory of the Fourier transform, which is of course the most fundamental tool in harmonic analysis… 8,006 more words

#### 245C, Notes 1: Interpolation of L^p spaces

In the previous two quarters, we have been focusing largely on the “soft” side of real analysis, which is primarily concerned with “qualitative” properties such as convergence, compactness, measurability, and so forth. 7,103 more words

#### 245B final; 245C course announcement

The 245B final can be found here. I am not posting solutions, but readers (both students and non-students) are welcome to discuss the final questions in the comments below. 112 more words