The Analysis in Hilbert space notes (MA40256) from 2017/18 are available here: MA40256.

Feel free to email me with any questions or queries, or for any typos: cc959@bath.ac.uk

Some more epsilon-delta examples (from tutorials and extra): examples.

Here are the solutions: example_solutions.

Here are some definitions of limits at infinity: limits_at_infinity.

There are also orders of growth on this sheet (big O and little o notation).

Here is the example of continuity of $f(x) = x^{-3}$ on $(0,2)$ from week 3: continuity_example.

Here is another example of continuity on an interval from week 4: continuity_example_2.

Here is a short introduction to open/closed sets and density in $\mathbb{R}$: open_closed

For a more indepth discussion, see MA20218 (Analysis 2A), in particular *metric spaces*.

The following proves Taylor's theorem with Lagrange and Cauchy error term: taylors_theorem_IVT.

This document also includes the statement of the *inverse function theorem*, as in PS6 H4.

Below are solutions to the "tutorial" questions on the problem sheets.

**Solutions 0**: solutions0.

**Solutions 1**: solutions1.

**Solutions 2**: solutions2.

**Solutions 3**: solutions3.

**Solutions 4**: solutions4.

**Solutions 5**: solutions5.

**Solutions 6**: solutions6.

**Solutions 7**: solutions7.