Mizar Commentary on Walter Rudin's Principles of Mathematical Analysis/Functions of Several Variables

Linear Transformations
9.1 Definition (a) Compare 1.36 Definitions (for vector spaces). Aside from that, REAL-NS n as a normed space is defined in REAL_NS1:def 4 and TOP-REAL n as a topological space is defined in EUCLID:def 8. (There is also REAL-US n (REAL_NS1:def 6).) Note that these are not a VectSp (compare NORMSTR (<tt>NORMSP_1</tt>), <tt>RLTopStruct</tt> (<tt>RLTOPSP1</tt>) and <tt>ModuleStr</tt> (<tt>VECTSP_1</tt>) in the second diagram here). <tt>RLSStruct</tt> (<tt>RLVECT_1</tt>) is incorporated into both <tt>NORMSTR</tt> and <tt>RLTopStruct</tt> and delivers most properties for both variants. RLS can be seen as a abbreviation of Real Linear Scalar. The <tt>RLSStruct</tt> variant of <tt>REAL n</tt> (<tt>EUCLID:def 1</tt>) would be <tt>RealVectSpace(Seg n)</tt> (<tt>FUNCSDOM:def 6</tt> and <tt>FINSEQ_1:def 1</tt>). Note that there is <tt>CLSStruct</tt> (<tt>CLVECT_1</tt>) which will not be discussed here. (b) A <tt>Linear_Combination</tt> of $$R^k$$ (<tt>RLVECT_2:def 3</tt>) is given as a function, sending the vectors to their associated scalars. <tt>Lin(S)</tt> (<tt>RLVECT_3:def 2</tt>) is the span of $$S$$. More generally, a <tt>Linear_Combination</tt> of a vector space is defined in <tt>VECTSP_6:def 1</tt> and <tt>Lin(S)</tt> in <tt>VECTSP_7:def 2</tt>. (c) Lineary independence is defined in <tt>RLVECT_3:def 1</tt> or more generally in <tt>VECTSP_7:def 1</tt>. (d) <tt>dim X</tt> is defined in <tt>RLVECT_5:def 2</tt> or more generally in <tt>VECTSP_9:def 1</tt>. The remark is given by <tt>RLVECT_5:32</tt> or <tt>VECTSP_9:29</tt>. (e) The <tt>Basis of X</tt> is defined in <tt>RLVECT_3:def 3</tt> or more generally in <tt>VECTSP_7</tt> via <tt>VECTSP_7:def 3</tt>. No reference for the remark or the standard basis.

9.2 Theorem No reference.

Corollary is <tt>EUCLID_7:46</tt> for <tt>RealVectSpace(Seg n)</tt>, <tt>RLAFFIN3:4</tt> for <tt>TOP-REAL n</tt> and <tt>EUCLID_7:51</tt> for <tt>REAL-US n</tt>, else no reference.

9.3 Theorem (a) No reference, but see <tt>RLVECT_5:29</tt>. (b) is given by <tt>existence</tt> property of the definition of a basis (see <tt>RLVECT_3:def 3</tt>). (c) is given by <tt>RLVECT_3:26</tt> or <tt>RLVECT_5:2</tt>.

9.4 Definition For <tt>VectSp</tt> $$A$$ is a <tt>linear-transformation</tt> (<tt>RANKNULL</tt>) if it is a <tt>additive</tt> (<tt>VECTSP_1:def 19</tt>), <tt>homogeneous</tt> (<tt>MOD_2:def 2</tt>) function. For <tt>RealLinearSpace</tt> $$A$$ is a <tt>LinearOperator</tt> (<tt>LOPBAN_1</tt>) if it is a <tt>additive</tt> (<tt>VECTSP_1:def 19</tt>), <tt>homogeneous</tt> (<tt>LOPBAN_1:def 5</tt>) function.

9.5 Theorem No reference.

9.6 Definitions (a) For <tt>RealLinearSpace</tt> $$L(X,Y)$$ is <tt>LinearOperators(X,Y)</tt> (<tt>LOPBAN_1:def 6</tt>), no reference fro <tt>VectSp</tt>. The addition of functions from $$X$$ to $$Y$$ is generally given by <tt>FuncAdd(X,Y)</tt> (<tt>LOPBAN_1:def 1</tt>, but also see <tt>FUNCT_3:def 7</tt>, <tt>FUNCOP_1:def 3</tt> and the redefinition of the latter in <tt>FUNCSDOM</tt> to understand the definition; or look at <tt>LOPBAN_1:1</tt>). Skalar multiplication of functions is only given for <tt>RealLinearSpace</tt> by <tt>FuncExtMult(X,Y)</tt> (<tt>LOPBAN_1:def 2</tt>), else no reference. (b) Composition was already defined <tt>RELAT_1:def 8</tt>. Additivity of the composition is clustered in <tt>GRCAT_1</tt> (see also <tt>GRCAT_1:7</tt>). Homogeneity of the composition is only given for the one defined in <tt>MOD_2:def 2</tt>, given by <tt>MOD_2:2</tt>, else no reference. However, the combination for <tt>RealLinearSpace</tt> is given by <tt>LOPBAN_2:1</tt>. (c) For a bounded operator $$A$$ between two <tt>RealNormSpace</tt> (<tt>NORMSP_1</tt>) $$X$$ and $$Y$$ the norm $$||A||$$ is given by <tt>BoundedLinearOperatorsNorm(X,Y).A</tt> (<tt>LOPBAN_1:def 13</tt>), before it is packed into <tt>R_NormSpace_of_BoundedLinearOperators</tt> (<tt>LOPBAN_1:def 14</tt>). The first inequality is given by <tt>LOPBAN_1:32</tt>, no reference for the second inequality. No reference for <tt>VectSp</tt> either.

9.7 Theorem (for <tt>VectSp</tt>) No reference.

9.7 Theorem (for <tt>RealNormSpace</tt>) (a) The first part is given implicitly by <tt>LOPBAN_1:27</tt>, the second one is given by <tt>LOPBAN_8:3</tt>. (b) First part is <tt>LOPBAN_1:37</tt>. The distance function is generally given by <tt>NORMSP_2:def 1</tt> and the generated metric space by <tt>NORMSP_2:def 2</tt>. (c) is <tt>LOPBAN_2:2</tt>.

9.8 Theorem No reference.

9.9 Matrices No reference.

Differentiation
9.10 Preliminaries The new viewpoint on differentiation is the one Mizar starts with, see 5.1 Definition and <tt>FDIFF_1</tt>.

9.11 Definition The notations are the same as for differentiation of real functions: $$f$$ is differentiable at $$x$$ means <tt>f is_differentiable_in x</tt> (<tt>NDIFF_1:def 6</tt>), $$f'(x)$$ is <tt>diff(f,x)</tt> (<tt>NDIFF_1:def 7</tt>). $$f$$ is differentiable on $$E$$ means <tt>f is_differentiable_on E</tt> (<tt>NDIFF_1:def 8</tt>), $$f'$$ is <tt>f`|E</tt> (<tt>NDIFF_1:def 9</tt>). See also <tt>PDIFF_1:def 7/8</tt>.

9.12 Theorem is given by the <tt>uniqueness</tt> property of <tt>NDIFF_1:def 7</tt>.

9.13 Remark (a) Basically given by <tt>NDIFF_1:47</tt>. (b) Given by the types of <tt>NDIFF_1:def 7</tt> and <tt>NDIFF_1:def 9</tt>. (c) is <tt>NDIFF_1:44/45</tt>. (d) Nothing to formalize.

9.14 Example No reference, but <tt>NDIFF_1:43</tt> is similar.

9.15 Theorem is <tt>NDIFF_2:13</tt>.

9.16 Partial derivates $$(D_j f_i)(x)$$ is <tt>partdiff(f,x,j,i)</tt> (<tt>PDIFF_1:def 16</tt>). The use of <tt>e_j</tt> and <tt>u_i</tt> is avoided by using <tt>reproj(j,x)</tt> (<tt>PDIFF_1:def 5/6</tt>) and <tt>Proj(i,m)</tt> (<tt>PDIFF_1:def 4</tt>) respectively.

9.17 Theorem No reference.

9.18 Example No reference.

9.19 Theorem No reference.

Corollary No reference.

9.20 Definition No reference.

9.21 Theorem is <tt>PDIFF_8:20-22</tt>.

The Contraction Principle
9.22 Definition is <tt>ALI2:def 1</tt> for <tt>MetrSpace</tt> and <tt>NFCONT_2:def 3</tt> for <tt>NORMSTR</tt>.

9.23 Theorem is <tt>ALI2:1</tt> for <tt>MetrSpace</tt> and <tt>NFCONT_2:14</tt> for <tt>RealBanachSpace</tt>.

The Inverse Function Theorem
9.24 Theorem No reference.

9.25 Theorem No reference.

The Implicit Function Theorem
9.26 Notation

9.27 Theorem

9.28 Theorem

9.29 Example No reference.