User:Barto

Please, post any opinions about the below on my talk page. There are some choices to be made, and I don't feel authorized to make them without anyone's consent.

Definitions
Good reference: http://www.math.illinois.edu/~ajh/ant/main.pdf (though it does not define asymptotic notations in full generality)


 * Definition:O Notation: include definitions for: (big $O$ as well as little $o$)
 * complex functions
 * relative estimates (=estimates in regions), real and complex
 * point estimates; real and complex, relative and absolute
 * non-uniform estimates (parameter-dependent)
 * Operations with $O$-estimates; e.g. define $f=g(O(h))$.


 * Definition:Asymptotically Equal: include definitions for: real, complex, relative/absolute, point-estimates, non-uniform


 * Mention majoration somewhere; Vinogradov's $\ll$ notation


 * Define $\asymp$ (Same order of magnitude relation) for: real, complex, relative/absolute, point-estimates, non-uniform

Properties

 * $O$ and Taylor-expansion of (real) analytic functions
 * Substitution of estimates
 * Integration of $O$ and $o$-estimates
 * Mention uniform-$O$ conventions
 * Differentiation of $O$-estimates

Infinite Products

 * Convergence and analyticity of analytic products
 * Logarithmic derivatives and analyticity

Convergence of Infinite Product of Complex Numbers
Let $P_n$ denote the $n$th partial product.

Suppose $(P_n)$ converges in $\C\setminus\{0\}$.

Then $(\log|P_n|)$ converges to $\log|z|$.

We have $\log|P_n|=\sum_{k=1}^n\Re\log(1+z_n)$.

It remains to show that $\sum_{k=1}^n\Im\log(1+z_n)\to\arg z+2k\pi$ for some $k\in\Z$.

Let $\theta=\arg z$ be the argument of $z$ and let $\theta_n=\arg P_n$ be the polar angle of $P_n$ in the half-open interval $(\theta-\pi,\theta+\pi]$, and let $k_n\in\Z$ be such that $|\theta+2k_n\pi-\theta_n|\leq\pi$.

By the convergence of $P_n$, $|\theta+2k_n\pi-\theta_n|\to0$.

We have $2\pi|k_n|\leq|\theta+2k_n\pi-\theta_n|+|\theta-\theta_n|\leq|\theta+2k_n\pi-\theta_n|+\pi$.

Because $k_n\in\Z$, $k_n=0$ for $n$ sufficiently large.

Modules

 * Make a page (or at least a transclusion) for Definition:Module Homomorphism. Note that there is Definition:Linear Transformation, Definition:R-Algebraic Structure Homomorphism and Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism where module isomorphisms are defined.
 * Consider renaming Definition:Free Module Indexed by Set to something less pedantic (name is used in some other articles; better decide quickly before there are more)

Direct Product and Direct Sum
and the naming of those subdefinitions should be consistent: Finite Case, General Case or something like that.
 * Make sure that the definition pages for direct products include a definition for arbitrary families. Ideally, there should be:
 * A definition for two components
 * A definition for a finite number of components
 * A definition for general families
 * (Optionally: a definition for countable families)


 * For direct sums, the definition for general families should suffice.


 * Be consistent in naming pages:
 * product vs. sum: product is the primary notion, sum is derived from it. It is thus impossible to define the direct sum without defining the direct product; unless when they coincide and allow for ambiguous naming.
 * External vs. internal: If nothing is specified, external has to be understood. IMO external can be removed from titles; it's unnecessarily pedantic; though I could live with it if it stays.
 * Word order: (less important) Direct Product of X's vs. X Direct Product? I'm in favor of Direct Product of X's, and accordingly for sum, internal or not internal.
 * The following table shows the jolly inconsistencies (no, I didn't misplace the entries):

Please, feel free to add missing entries. These are the only definitions I found so far (not including transclusions). Some definitions are still to be extracted from a theorem that's being refactored.

Dimension

 * Definition:Basis (Linear Algebra): refactor definitions; show equivalence; add definitino using Definition:Free Module on Set
 * Definition:Dimension (Linear Algebra): mention that it is well-defined, with reference
 * Definition:Ordered Basis
 * Unitary R-Modules with n-Element Bases Isomorphic
 * Isomorphism from R^n via n-Term Sequence
 * Unique Representation by Ordered Basis

Support

 * Elements with Support in Ideal form Submagma of Direct Product: fix notion of ideal and link
 * Elements of Finite Support form Submagma of Direct Product same
 * Support of Product is Contained in Union of Supports How to organize?

Polynomials
Specifically, Definition:Polynomial (Abstract Algebra)


 * Many definitions are nothing but special cases: once we have defined a polynomial over a ring, there's no need to define it once more over an integral domain. Not only is this superfluous, it's also wrong (albeit in a very mild sense in this case) to define something twice. Maybe we can leave a note for polynomials over fields, but it should be emphasized that it is defined as the polynomial over the field considered as ring.

Solution: rename those definition pages to reflect what they do define: "polynomials in elements" or "polynomial elements" (I don't know what they're usually called), and not polynomial expressions, whatever that might mean
 * Some definitions seem to define polynomials over a ring $R$ as polynomials "in an element" $x\in R$. There are two problems with this:
 * Some notions, such as Definition:Degree of Polynomial/Ring are ill-defined. Probably there are more such issues, such as leading coefficient etc
 * It gives no way to introduce a variable $X$; that is, to construct an actual polynomial ring because those polynomials are not new elements that were not already in the ring. Said otherwise, one has to invoke two definitions to make an actual polynomial: one introducing a variable and making a ring out of that, one to consider polynomials.


 * I've never heard Definition:Ring of Polynomial Forms before visiting ProofWiki. I'd rather call it simply Definition:Ring of Polynomials, which is unfortunately already occupied (redirect page) by a much less common notion.


 * Add a definition of polynomials ("polynomial forms", if you wish) using Definition:Monoid Ring


 * Add a definition of polynomial functions (over ring $R$) using the evaluation morphism in the identity to the ring of functions $R\to R$ (with pointwise operations)

Modules

 * Address the cardinality of a basis of a module
 * Define modules and unitary modules using Definition:Endomorphism Ring of Abelian Group, see Morphism to Endomorphism Ring Defines Module
 * Determining Definition:Multilinear Mapping using basises, use this in Definition:Monoid Ring
 * Correspondence between Definition:Multilinear Mapping and linear maps to Definition:Module of Multilinear Mappings.

Definition-level

 * Support of Product is Contained in Union of Supports


 * Free Module Indexed by Set is Free
 * Canonical Basis of Free Module Indexed by Set is Basis
 * Endomorphism Ring of Abelian Group is Ring
 * Module of Homomorphisms between Modules is Module
 * Monoid Ring is Ring


 * Universal Property of Free Modules
 * Universal Property of Direct Product of Modules
 * Universal Property of Direct Sum of Modules
 * Universal Property of Monoid Ring

Spaces of Morphisms

 * Definition:Dual Module
 * Definition:Group of Homomorphisms Between Abelian Groups
 * Definition:Endomorphism Ring of Abelian Group
 * Definition:Module of Homomorphisms Between Modules
 * Definition:Module of Multilinear Mappings
 * Definition:Endomorphism Ring of Module
 * Note that there is already a general Definition:Group of Automorphisms for algebraic structures with one operation.

Long Term Projects

 * Disambiguate between left and right modules. Does this mean that every occurrence of "module" should be amended?