One way to study a general class of mathematical objects is to embed them into a more structured class of mathematical objects; for instance, one could study manifolds by embedding them into Euclidean spaces. 1,457 more words

## Tags » 245B - Real Analysis

#### 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

#### 245B, Notes 12: Continuous functions on locally compact Hausdorff spaces

A key theme in real analysis is that of studying general functions or by first approximating them by “simpler” or “nicer” functions. But the precise class of “simple” or “nice” functions may vary from context to context. 7,446 more words

#### Tricks Wiki: Give yourself an epsilon of room

Today I’d like to discuss (in the Tricks Wiki format) a fundamental trick in “soft” analysis, sometimes known as the “limiting argument” or “epsilon regularisation argument”. 2,897 more words

#### 245B, Notes 11: The strong and weak topologies

A normed vector space automatically generates a topology, known as the *norm topology* or *strong topology* on , generated by the open balls . A sequence in such a space… 5,117 more words

#### 245B, Notes 10: Compactness in topological spaces

One of the most useful concepts for analysis that arise from topology and metric spaces is the concept of compactness; recall that a space is compact if every open cover of has a finite subcover, or equivalently if any collection of closed sets with the… 5,311 more words

#### 245B, Notes 9: The Baire category theorem and its Banach space consequences

The notion of what it means for a subset E of a space X to be “small” varies from context to context. For instance, in measure theory, when is a measure space, one useful notion of a “small” set is that of a null set: a set E of measure zero (or at least contained in a set of measure zero). 5,182 more words