Mizar Commentary on Walter Rudin's Principles of Mathematical Analysis/Sequences and Series of Functions

Note that sequences of functions can be seen as sequences over a normed space, which will mostly be left untouched in this chapter.

Discussion of main problem
7.1 Definition Let $$F$$ denote $$\{f_n\}$$. $$F$$ being pointwise convergent is described by F is_point_conv_on E (SEQFUNC:20, see also SEQFUNC:def 10). Then $$\lim_{n\to\infty}f_n$$ is given by lim(F,E) (SEQFUNC:def 13). No reference for series.

7.2 Example No reference.

7.3 Example No reference.

7.4 Example No reference.

7.5 Example No reference.

7.6 Example No reference.

Uniform Convergence
7.7 Definition $$F$$ being uniformly convergent on $$E$$ is F is_unif_conv_on E (SEQFUNC:def 12) or F is_uniformly_convergent_to f (<tt>MESFUN10:def 2</tt>. That uniform convergence implies pointwise convergence is <tt>SEQFUNC:22</tt> or <tt>MESFUN10:20</tt>. No reference for function series.

7.8 Theorem No reference.

7.9 Theorem No reference.

7.10 Theorem No reference.

Uniform Convergence and Continuity
7.11 Theorem No direct reference, but implicitly given in proof of <tt>TIETZE:20</tt>.

7.12 Theorem is <tt>TIETZE:20</tt>.

7.13 Theorem No reference.

7.14 Definition No reference.

7.15 Theorem No reference.

Uniform Convergence and Integration
7.16 Theorem is <tt>INTEGRA7:31</tt> or <tt>MESFUN10:21</tt>.

Corollary No reference.

Uniform Convergence and Differentiation
7.17 Theorem No reference. #TODO show that $$f'(x)=\lim_{n\to\infty}f_n(x)$$

7.18 Theorem No reference. #TODO existence of continuous nowhere differentiable function

Equicontinuous Families of Functions
7.19 Definition No reference for pointwise bounded, but for <tt>F is uniformly_bounded</tt> (<tt>MESFUN10:def 1</tt>).

7.20 Example No reference.

7.21 Example No reference.

7.22 Definition No reference.

7.23 Theorem No reference.

7.24 Theorem No reference.

7.25 Theorem No reference.

The Stone-Weierstrass Theorem
7.26 Theorem No reference #TODO the Stone-Weierstrass theorem

7.27 Corollary No reference.

7.28 Definition The algebra of complex functions on a domain $$E$$ is <tt>CAlgebra(E)</tt> (<tt>CFUNCDOM:def 8</tt>), the real analogon of that is <tt>RAlgebra(E)</tt> (<tt>FUNCSDOM:def 9</tt>). A general algebra structure is found in <tt>POLYALG</tt> (not to be confused with the universal algebra from <tt>UNIALG_1</tt>). Subalgebras for complex algebras are defined in <tt>CC0SP1:def 1</tt>, for real algebras in <tt>C0SP1:def 9</tt>. No reference for uniformly closed or uniformly closure.

7.29 Theorem No reference.

7.30 Definition No reference.

7.31 Theorem No reference.

7.32 Theorem No reference.

7.33 Theorem No reference.

Exercises
1. No reference. 2. No reference. 3. No reference. 4. No reference. 5. No reference. 6. No reference. 7. No reference. 8. No reference. 9. No reference. 10. No reference. 11. No reference. 12. No reference. 13. No reference. 14. No reference. 15. No reference. 16. No reference. 17. No reference. 18. No reference. 19. No reference. 20. No reference. 21. No reference. 22. No reference. 23. No reference. 24. No reference. 25. No reference. 26. No reference.