# 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 (MESFUN10:def 2. That uniform convergence implies pointwise convergence is SEQFUNC:22 or MESFUN10:20. 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 TIETZE:20.

7.12 Theorem is TIETZE:20.

7.13 Theorem No reference.

7.14 Definition
No reference.

7.15 Theorem No reference.

## Uniform Convergence and Integration

7.16 Theorem is INTEGRA7:31 or MESFUN10:21.

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 F is uniformly_bounded (MESFUN10:def 1).

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 CAlgebra(E) (CFUNCDOM:def 8), the real analogon of that is RAlgebra(E) (FUNCSDOM:def 9). A general algebra structure is found in POLYALG (not to be confused with the universal algebra from UNIALG_1). Subalgebras for complex algebras are defined in CC0SP1:def 1, for real algebras in C0SP1:def 9. 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

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