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• Nabla
  ...partial(\ )}{\partial x}+\mathbf{j}\frac{\partial(\ )}{\partial y}+\mathbf{k}\frac{\partial(\ )}{\partial z}</math> ...ok (Vector Analysis and Cartesian Tensors - Third Edition - D E Borune & P C Kendall - which is a good book) distinguishbetween the <math>\nabla</math>s
  1 KB (245 words) - 18:35, 13 February 2015
• Subspace topology
  {{Requires proof|grade=C|msg=Really easy, hence low importance|easy=true}} {{Requires proof|grade=C|msg=Really easy, hence low importance|easy=true}}
  6 KB (1,146 words) - 23:04, 25 September 2016
• Function
  ...is a space with some useful property, this always means {{M|f:A\rightarrow C}}, for example: ...gical space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} then we may write:
  4 KB (659 words) - 13:01, 19 February 2016
• Norm
  ...'obvious' as if the image of {{M|\Vert\cdot\Vert}} could be in {{M|\mathbb{C} }} then the {{M|\Vert x\Vert\ge 0}} would make no sense. What ordering wou ...langle\cdot,\cdot\rangle:V\times V\rightarrow(\mathbb{R}\text{ or }\mathbb{C})}} induces a ''norm'' given by:
  6 KB (1,026 words) - 20:33, 9 April 2017
• Index of notation
  ...to {{M|\mathbb{R} }} - this is unlikely to be given any other way because "C" is for continuous. | {{M|\mathbb{S}^n}}, {{M|l_2}}, {{M|\mathcal{C}[a,b]}}
  9 KB (1,490 words) - 06:13, 1 January 2017
• Basis and coordinates
  ...one, <math>\{b_1,...,b_n\}</math>, a point {{M|p}} is given by <m>\sum^n_{k=1}a_ib_i</m> and it is said to have coordinates <math>(a_1,...,a_n)</math> ...math> and it is easy to show that this is linear. Let us call this map {{M|K}} and define it as follows:
  9 KB (1,525 words) - 16:30, 23 August 2015
• Smooth manifold
  ...manifolds - John M Lee - Second Edition</ref> {{M|f:M\rightarrow\mathbb{R}^k}} that satisfies: ...such that {{M|f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k }} is [[Smooth|smooth]] in the usual sense, of having continuous partial de
  3 KB (413 words) - 21:09, 12 April 2015
• Smooth function
  ...f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\rightarrow\mathbb{R}\in C^\infty]}} Note that given an {{M|f:M\rightarrow\mathbb{R}^k}} this is actually just a set of functions, {{M|f_1,\cdots,f_k}} where {{M|
  3 KB (560 words) - 16:16, 14 April 2015
• Addition of vector spaces
  '''Operations:''' (given {{M|u_i,v_i\in V_i}} and {{M|c}} is a scalar in {{M|F}}) * <math>c(v_1,\cdots,v_n)=(cv_1,\cdots,cv_n)</math>
  4 KB (804 words) - 18:02, 18 March 2016
• Index of norms and absolute values
  | <math>\|\cdot\|_{C^k}</math> | <math>\|f\|_{C^k}</math>
  1 KB (207 words) - 09:16, 9 June 2015
• Bounded set
  ...\forall a,b\in A[d(a,b)<C]}} - where {{M|C}} is real<ref group="Note">{{M|C\in\mathbb{R}_{\ge 0} }} should do as {{M|0}} could be a bound, I suppose on ...x\in A}} (where {{M|1=x=(x_1,\cdots,x_n)}}) we have {{M|\vert x_i\vert\le K}} for {{M|i\in\{1,\cdots,n\} }}
  2 KB (409 words) - 23:31, 29 October 2016
• Index of spaces
  ...exed as "num" (notice the lower-case) so a space like {{M|C^k}} is under {{C|C_num}}. We do subscripts first, so {{M|A_i^2}} would be under {{C|A _num ^num:2}}
  3 KB (612 words) - 21:06, 29 February 2016
• Borel sigma-algebra
  ! {{M|1=\mathcal{B}^n=\sigma(\mathcal{C})}} - closed<ref name="MIM"/> ...sigma(\mathcal{O})}} and {{M|1=\sigma(\mathcal{O})\subseteq\sigma(\mathcal{C})}} - see '''Claim 1'''
  5 KB (854 words) - 09:25, 6 August 2015
• Classes of continuously differentiable functions
  ...|C^k}} on {{M|U}}, {{M|C^k(U)}} for {{M|U\subseteq\mathbb{R}^n}} and {{M|C^k(\mathbb{R}^n)}} such. ...U\subseteq\mathbb{R}^n}} (where {{M|U}} is [[Open set|open]]) and some {{M|k\ge 0}}, a function of the form:
  3 KB (632 words) - 20:32, 16 October 2015
• Class of smooth real-valued functions on R-n
  * {{M|C^\infty(\mathbb{R}^n)}} {{Note|The conventions concerning the {{M|C^k}} notation are addressed on the page: ''[[Classes of continuously different
  2 KB (259 words) - 23:41, 21 October 2015
• Sandbox
  I \ar[r]^f \ar[d] \ar@{.>}[dr] & J \ar[r]^g & K \\ C \ar@(dr,dl)@2{_{(}->}[r] _a = "a" & D \ar@{_{(}-^{)} }[r]_b = "b" & Y
  695 B (132 words) - 22:15, 26 October 2015
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/1 implies 2
  :* So {{MM|1=\exists C>0\ \forall N\in\mathbb{N}\ \exists n\in\mathbb{N}[n>N\wedge\Vert L(x_n-p)\V ...at {{M|d(x_n,x)>\epsilon}}, we shall later call such an {{M|\epsilon}} {{M|C}} and construct a subsequence out of the {{M|n}}s
  5 KB (1,064 words) - 02:24, 28 February 2016
• Notes:Boundary operator
  * {{M|1=\partial_p:C_p(K)\rightarrow C_{p-1}(K)}} given by {{M|1=\partial_p\sigma:=\partial_p[v_0,\ldots,v_p]=\sum^p_{i=0} ...is a function, {{M|c}}, from the set of oriented {{M|p}}-simplicies of {{M|K}} to {{M|\mathbb{Z} }} such that:
  1 KB (257 words) - 00:29, 8 May 2016
• Notes:Differential (manifolds)
  * '''Smoothness of a map ({{AKA}}: {{M|C^\infty}}''' - a map, {{M|f:U\subseteq\mathbb{R}^n\rightarrow V\subseteq\mat * '''[[Derivation]]''' - a map, {{M|\omega:C^\infty(M)\rightarrow\mathbb{R} }} that is [[linear map|linear]] and satisfi
  4 KB (716 words) - 14:24, 16 May 2016
• Notes:Mean value theorem
  ** {{M|\exists c\in(a,b)}} such that {{MM|1=f'(c)=\frac{f(a)-f(b)}{b-a} }} ...row\mathbb{R} \in}} [[Classes of continuously differentiable functions|{{M|C^1}} - the class of functions with continuous partial derivatives]]. Let {{M
  3 KB (529 words) - 08:07, 4 June 2016
• Task:Continuity types
  # Metric space version of {{C|1)}} (for {{M|(X,d_1)}} and {{M|(Y,d_2)}} being [[metric space|metric space ...M|f:X\rightarrow Y}} (for [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}) we have "topological continuity at a point":
  3 KB (668 words) - 22:38, 4 August 2016
• Notes:Dual basis
  ...a ''finite dimensional'' [[vector space]] over the [[field]], {{M|\mathcal{K} }}, suppose it has dimension {{M|n\in\mathbb{N} }}. ...set]] consisting of all [[function|functions]], {{M|f:V\rightarrow\mathcal{K} }} which are [[linear map|linear maps]].
  5 KB (1,020 words) - 08:43, 12 August 2016
• Topological vector space
  ...\eq\mathbb{R} }}, or [[the complex numbers]], so {{M|\mathbb{K}:\eq\mathbb{C} }} and let {{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.| * {{M|(X,\mathcal{J},\mathbb{K})}}<ref group="Note">This tuple doesn't really matter, nor does the order.
  2 KB (383 words) - 14:03, 16 February 2017
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary [ ** Let {{M|C^0(X,Y)}} denote the [[set]] of all [[continuous maps]] of the form {{M|(:X\
  2 KB (272 words) - 23:37, 14 October 2016
• Notes:Homotopy terminology/Terminology
  ...]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. Let {{M|C^0(X,Y)}} denote the set of all [[continuous maps]] of the form {{M|(:X\righ # '''Homotopic - ''' a relation on maps {{M|f,g\in C^0(X,Y)}}. We write {{M|f\simeq g\ (\text{rel }A)}} if there exists a homoto
  900 B (184 words) - 14:40, 16 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces/proof
  # For all {{M|f\in C^0(X,Y)}} that {{M|f\simeq f\ (\text{rel }A)}}, symbolically: #* [[Reflexive]]: {{M|1=\forall f\in C^0(X,Y)[\homo{f}{f}]}}
  3 KB (533 words) - 07:33, 18 September 2016
• Characteristic property of the disjoint union topology/Statement
  {{Stub page|msg=A rewrite, while not urgent, would be nice|grade=C}} ...\alpha\in I} }} be a collection of [[topological spaces]] and let {{Top.|Y|K}} be another topological space]]. We denote by {{M|\coprod_{\alpha\in I}X_\
  1 KB (238 words) - 20:05, 25 September 2016
• Doctrine:Homotopy terminology
  * Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] ...ans continuous with continuous first and second derivatives, so forth, {{M|C^\infty}} means [[smooth]].
  3 KB (535 words) - 09:01, 31 October 2016
• Closed map
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]] (no ...lural|closed set|s}} of {{Top.|X|J}} are [[closed set|closed]] in {{Top.|Y|K}}
  1 KB (246 words) - 19:59, 26 September 2016
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
  ...y|discrete topological space]] with {{M|1=Y:=\{0,1\} }} and {{M|1=\mathcal{K}:=\mathcal{P}(Y)}}<ref group="Note">Note: {{M|1=\mathcal{P}(Y)=\mathcal{P}( {{Requires proof|grade=C|easy=true|msg=Easy proof to do, exercise on page 87 in Lee's top. manifolds
  1 KB (172 words) - 23:12, 30 September 2016
• The image of a compact set is compact
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|A\in\mathcal{P}(X)}} be an arbitrary [ {{Requires proof|grade=C|msg=Proof isn't that important as it is easy and routine.
  2 KB (332 words) - 17:20, 18 December 2016
• Homeomorphic
  Two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}}, are said to be ''homeomorphic'' if there exists a [[homeomorphism]] betw ...he morphisms (which are continuous maps) between {{Top.|X|J}} and {{Top.|Y|K}}.
  883 B (132 words) - 11:52, 8 October 2016
• If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[surjectiv {{Requires proof|grade=C|msg=This could be more formal}}
  2 KB (264 words) - 22:32, 9 October 2016
• A map is continuous if and only if the pre-image of every closed set is closed
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[mapping]] ...{M|1=\forall E\in C(\mathcal{K})[f^{-1}(E)\in C(\mathcal{J})]}} (where {{M|C(\mathcal{H})}} denotes the ''[[set of all closed sets|set of all]]'' [[clos
  2 KB (378 words) - 01:39, 14 October 2016
• Exercises:Measure Theory - 2016 - 1/Section B/Problem 1
  #*** Define {{M|1=k:=\text{Max}(\{i,j\})}} #***** {{M|\forall C,D\in\mathcal{A}_k[C\cup D\in\mathcal{A}_k]}}
  10 KB (1,844 words) - 14:09, 23 October 2016
• The set of continuous functions between topological spaces
  ...spect to the {{plural|topolog|y|ies}}: {{M|\mathcal{J} }} and {{M|\mathcal{K} }}. * {{M|\big(f\in C(X,Y)\big)\iff\big(f:X\rightarrow Y\text{ is a continuous function}\big)}}
  1 KB (235 words) - 05:02, 3 November 2016
• Homotopy concatenation
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Let {{M|H_1:X\times I\rightarrow Y}} and {{M|H {{Requires proof|grade=C|msg=I've basically done it. See note above}}
  2 KB (260 words) - 05:09, 6 November 2016
• Homotopy invariance of loop concatenation
  ...ga(X,b)|{{M|\Omega(X,b)}} is a [[subset of|subset]] of {{C(I,X)}}, and {{M|C(I,X)}} is the set of all {{link|path|topology|s}} in {{M|X}}. {{M|\Omega(X,
  3 KB (462 words) - 09:21, 6 November 2016
• The vector space of all linear maps between two spaces
  ...athbb{K})}} and {{M|(V,\mathbb{K})}} be [[vector spaces]] over {{M|\mathbb{K} }}. We define: * '''Claim 1: ''' {{M|L(U,V)}} is a [[vector space]] over {{M|\mathbb{K} }} in its own right
  2 KB (400 words) - 21:16, 17 November 2016
• A monotonically increasing sequence bounded above converges
  {{Stub page|grade=C|msg=Good enough for now, routine first year work anyway}} .... {{M|\exists b\in\mathbb{R}\exists K\in\mathbb{N}\forall n\in\mathbb{N}[n>K\implies a_n\le b]}} - {{M|b}} is the bound) then:
  3 KB (493 words) - 07:21, 23 November 2016
• Notes:Richard Sharp question
  ...rt_\infty \frac{1}{n^2\delta^2}\sum_{k\eq 0}^n(k-nx)^2\ {}^nC_kx^k(1-x)^{n-k} }}</span> ...ert_\infty \sum_{k\eq 0}^n\frac{(k-nx)^2}{n^2\delta^2}\ {}^nC_kx^k(1-x)^{n-k} }}
  2 KB (424 words) - 21:03, 25 November 2016
• Weierstrass approximation theorem
  ** {{M|\forall f\in C([0,1],\mathbb{R})\forall\epsilon>0\exists n\in\mathbb{N}[\Vert f-B_N(f)\Ver Let {{M|C([a,b],\mathbb{R})}} denote the [[vector space of continuous functions from
  8 KB (1,610 words) - 08:17, 28 December 2016
• Transposition (group theory)
  * Demote to grade {{C|D}} once more content is added}} Let {{M|S_k}} denote the [[symmetric group]] on {{M|k\in\mathbb{N} }} symbols. Then:
  778 B (136 words) - 10:28, 30 November 2016
• Notes:An introduction to manifolds - Loring W. Tu
  ====1.1: {{M|C^\infty}} vs Analytic functions==== ...eq\lim_{t\rightarrow 0}\left(\frac{f(c(t))-f(p)}{t}\right)\eq\frac{d}{dt}f(c(t))\Big\vert_{t\eq 0} }}
  5 KB (910 words) - 14:07, 1 December 2016
• Characteristic property of the tensor product/Statement
  : '''Notice: ''' this page is supposed to be transcluded, use {{C|1=full=true}} to show claims and extra things ...athbb{F} }} be a [[field]] and let {{M|\big((V_i,\mathbb{F})\big)_{i\eq 1}^k}} be a family of ''[[dimension (vector space)|finite dimensional]]'' [[vect
  2 KB (268 words) - 22:07, 20 December 2016
• Characteristic property of the tensor product
  * Let {{M|i\in\{1,\ldots,k\} }} be given * Since {{M|i\in \{1,\ldots,k\} }} was arbitrary, we see it is [[linear map|linear]] in each {{M|i}} - th
  3 KB (626 words) - 23:51, 3 December 2016
• Basis for the tensor product
  ...{\alpha\eq 1}^m \lambda_\alpha (v_{\alpha,1}\otimes\cdots\otimes v_{\alpha,k})}} for some {{M|\lambda_\alpha\in\mathbb{F} }} and {{M|v_{\alpha,i}\in V_i ...}\right)\otimes\cdots\otimes\left(\sum_{i_k\eq 1}^{n_k}v_{\alpha,k,i_k}e^{(k)}_{i_k}\right)\right)}}
  3 KB (489 words) - 23:56, 6 December 2016
• The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|\varphi:X\rightarrow Y}} be a [[contin {{Requires proof|grade=C|msg=It remains to be shown that {{M|H'}} is a [[homotopy]] between {{M|(\va
  2 KB (282 words) - 22:38, 12 December 2016
• Tensor product of tensors
  ...{M|\ell}} respectively", then we can define the tensor product, a rank {{M|k+\ell}} tensor, as follows{{rAOMJRM}}: ...''' {{M|(cf)\otimes g\eq c(f\otimes g)\eq f\otimes(cg)}} - for scalar {{M|c\in\mathbb{F} }}
  3 KB (497 words) - 21:58, 22 December 2016
• Index of notation for sets of continuous maps/Index
  ...C(X,Y)}}]] - for [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}, {{M|C(X,Y)}} is the [[set]] of all [[continuous maps]] between them. # [[C(I,X)|{{M|C(I,X)}}]] - {{M|I:\eq[0,1]\subset\mathbb{R} }}, set of all {{link|path|topol
  2 KB (463 words) - 06:20, 1 January 2017
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.2
  ...PQRSTUVWXYZ} }}. I notice that {{M|\sf{G} }} here is homeomorphic to a {{M|C}}, so I have included {{M|\underline{\text{G} } }}, this represents {{M|G}} * I also include {{M|\mathcal{Z} }} representing a {{M|\sf{Z} }} with a {{C|-}} through the middle, again due to how common this form is
  17 KB (3,132 words) - 12:03, 18 January 2017
• Notes:Stone-Weierstrass theorem
  ...is strictly [[corollary]] to {{M|A}}, that is {{C|A{{M|\implies}}B}} but {{C|B ''(does not imply)'' A}} ...|)}} be given^Important:<ref group="Note">There are a lot of {{C|K}}s in play here. As per ''[[Doctrine:Notation for sets of continuous maps]]
  4 KB (791 words) - 13:29, 27 January 2017
• Exercises:Saul - Algebraic Topology - 3/Exercise 3.2
  ...can represent the path by the formal linear combination: {{M|\sum_{i\eq 1}^k p_i}}. ...the Abelian groups that usually have a {{M|C}} in them and how he used {{M|C}}.
  13 KB (2,312 words) - 06:33, 1 February 2017
• Notes:Delta complex/Formal attempt
  ...complex]] in the back of your mind, and a simplex as being like {{M|\{a,b,c\} }} for a triangle and such. * Let {{M|\#(n):\eq\{1,\ldots,n\}\subset\mathbb{N} }} - I did want to use {{M|C(n)}} for "count" or "consecutive" but given the context that'd be a poor ch
  5 KB (966 words) - 14:36, 6 February 2017
• Convex set
  {{Stub page|grade=C|msg=Needs a reference and tidying up! ...{M\mathbb{C} }} and let {{M|C\in\mathcal{P}(X)}} be given. Then we say {{M|C}} is ''convex'' if:
  799 B (143 words) - 11:27, 9 February 2017
• Convex
  ...r a [[vector space]] {{M|(X,\mathbb{K})}} a [[subset of]] {{M|X}}, say {{M|C\in\mathcal{P}(X)}} is convex if ** {{M|\forall x,y\in C\forall t\in[0,1]\subset\mathbb{R}[x+t(y-x)\in C]}} - the line between any two points in the set is also in the set
  532 B (92 words) - 14:50, 9 February 2017
• Notes:Functional Analysis II
  ...als]], {{M|\mathbb{R} }}, or the field of [[complex numbers]], {{M|\mathbb{C} }} *# {{M|\forall x\in X\forall\lambda\in\mathbb{K}[N(\lambda x)\eq\vert\lambda\vert N(x)]}} {{XXX|Positive definiteness or so
  4 KB (818 words) - 12:00, 9 February 2017
• Doctrine:K (field)
  {{DISPLAYTITLE:Doctrine:{{M|\mathbb{K} }} (field)}} ...\eq\mathbb{R} }}, or [[the complex numbers]], so {{M|\mathbb{K}:\eq\mathbb{C} }}
  622 B (94 words) - 18:30, 16 February 2017
• Doctrine:K (topological space)
  {{DISPLAYTITLE:Doctrine:{{M|K}} (topological space)}} {{M|K}} shall denote the (underlying set) of any [[topological space]] which is {
  518 B (76 words) - 18:36, 16 February 2017
• Doctrine:K
  ...ither [[the reals|{{M|\mathbb{R} }}]] or [[the complex numbers|{{M|\mathbb{C} }}]] ...l space)]] - any ''{{link|compact|topology}}'' [[topological space]], {{M|(K,\mathcal{J})}} where {{M|\mathcal{J} }} is the associated [[topology]]
  617 B (92 words) - 18:42, 16 February 2017
• Notes:Ell^p(C) is complete for p between one and positive infinity inclusive
  ...\subseteq\overline{\mathbb{R} } }} the space [[ell^p(C)|{{M|\ell^p(\mathbb{C})}}]] is [[complete metric space|complete]]. * Let {{M|(\mathbf{x}_n)_{n\in\mathbb{N} }\subseteq\ell^p(\mathbb{C})}} be given
  4 KB (664 words) - 18:56, 22 February 2017
• Evenly covered by a continuous map
  Let {{Top.|C|K}} and {{Top.|X|J}} be [[topological spaces]] and let {{M|f:C\rightarrow x}} be a [[continuous map]]. Let {{M|U\in\mathcal{J} }} be given ** {{MM|\exists \{V_\alpha\}_{\alpha\in I}\subseteq\mathcal{K} }} - there exists an arbitrary collection of open sets - such that:
  2 KB (394 words) - 21:54, 24 February 2017
• Notes:Covering spaces
  Let {{M|p:E\rightarrow X}} be a covering map, let {{Top.|Y|K}} be a [[topological space]] and let {{M|f:Y\rightarrow X}} be a [[continuo ...{{M|p:E\rightarrow X}} be a covering map, let {{M|\gamma\in}}[[C(I,X)|{{M|C([0,1],X)}}]] (a {{link|path|topology}}), then as {{M|p}} is a covering map
  3 KB (658 words) - 19:20, 25 February 2017
• Borel sigma-algebra of the real line
  # {{M|\{(c,+\infty)\ \vert\ c\in\mathbb{M}\} }}{{rMIAMRLS}} # {{M|\{[c,d)\ \vert\ c,d\in\mathbb{M}\} }}{{rMIAMRLS}}
  4 KB (712 words) - 15:48, 27 February 2017
• Unique lifting property
  ...h {{link|covering map|topology}} {{M|p:E\rightarrow X}}). Suppose {{Top.|Y|K}} is a [[connected topological space]] and {{M|f:Y\rightarrow X}} is a [[co {{Requires work|grade=C|msg={{Warning|What follows is VERY messy.}} I was distracted when writing i
  13 KB (2,510 words) - 16:23, 2 March 2017
• A sequence consisting of the nth terms of the sequences in a Cauchy sequence of elements in any little-L space is itself a Cauchy sequence of complex numbers
  ...o elements in {{M|\ell^p}} are {{M|(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C} }} such that certain properties hold. Let {{M|\big((x_n^k)_{n\in\mathbb{N} }\big)_{k\in\mathbb{N} } \subseteq\ell^p}} be a [[Cauchy sequence]]
  1 KB (238 words) - 17:52, 18 March 2017
• If two charts are smoothly compatible with an atlas then they are smothly compatible with each other
  Recall that to be considered smooth or {{M|C^\infty}} they must be smooth at each point in the domain, we will show {{M| {{Theorem Of|Manifolds|Smooth Manifolds|C-k manifolds|Topology}}
  6 KB (1,182 words) - 13:38, 1 April 2017
• Example:Smooth map between the real line and the circle (considered as smooth manifolds)
  * Let {{M|p\in\mathbb{R} }} be given, we will show that {{M|F}} is {{M|C^\infty}}/smooth here, since this is arbitrary, smooth everywhere. ...eq \varphi_+\circ F:\bigcup_{k\in\mathbb{Z} }\underbrace{\big(2\pi k, 2\pi k+\pi\big)}_{\subseteq\mathbb{R} }\rightarrow \underbrace{I:\eq(-1,1)}_{\subs
  4 KB (757 words) - 13:25, 2 April 2017
• If an inner product is non-zero then both arguments are non-zero
  Let {{M|((X,}}[[K (field)|{{M|\mathbb{K} }}]]{{M|),\langle\cdot,\cdot\rangle)}} be an [[inner product space]], then {{Requires proof|grade=C|msg=Easy proof|easy=true}}
  1 KB (181 words) - 23:40, 7 April 2017
• Given a Hilbert space and a non-empty, closed and convex subset then for each point in the space there is a closest point in the subset
  ...a field means {{M|\mathbb{K}:\eq\mathbb{R} }} or {{M|\mathbb{K}:\eq\mathbb{C} }}, the reals or the complex number fields only</ref> considered as a [[ve ...orall a,b\in A\forall\lambda\in[0,1]\subset\mathbb{R}\big(\subseteq\mathbb{K}\big)[a+\lambda(b-a)\in A]}}
  3 KB (592 words) - 00:52, 7 April 2017
• K (field)
  ...|\mathbb{K} }} means when encountered as a [[field]] (eg if {{M|(X,\mathbb{K})}} is a [[vector space]] - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 03 ...ers]], {{M|\mathbb{R} }}, or the [[field of complex numbers]], {{M|\mathbb{C} }}.
  2 KB (323 words) - 03:54, 8 April 2017
• The norm of a space is a uniformly continuous map with respect to the topology it induces
  {{Stub page|grade=C|msg=Check over before removing this, ensure it's linked to, maybe add "unif Let {{M|((X,}}[[K (field)|{{M|\mathbb{K} }}]]{{M|),\Vert\cdot\Vert)}} be a [[normed space]], we claim that the [[no
  4 KB (687 words) - 20:59, 9 April 2017
• Homotopy equivalent topological spaces
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], we say {{M|X}} is ''homotopy equivalent'' to { * {{M|\exists f\in}}[[C(X,Y)|{{M|C(X,Y)}}]]{{M|\exists g\in C(Y,X)\big[(g\circ f\simeq }}[[Identity map|{{M|\text{Id}_X}}]]{{M|)\wedge(g\
  3 KB (596 words) - 21:13, 24 April 2017
• Square lemma (of homotopic paths)
  * {{M|k:[0,1]\rightarrow X}} by {{M|k(t):\eq F(t,1)}} * {{M|(f*g)\simeq(h*k)\ (\text{rel }\{0,1\})}}
  2 KB (435 words) - 14:51, 25 April 2017
• Experiment:The wx wrapping function
  ...wxGraph'' project folder, and {{C|wxStaticWrappedText}} (my class, not a {{C|wx}} class) ...{{C|wxString}} to the label of a {{C|wxStaticText}}, we then tell it to {{C|Wrap}} passing the usable width as a parameter, the wrap function takes the
  2 KB (371 words) - 16:38, 23 April 2017
• List of common homology groups
  * {{M|H_k(\mathbb{S}^2)\cong 0}} for {{M|k\ge 3}} \bullet \ar@{}@(l,d)[]^{v_1} \ar[ur]^A_B_(.75)c^(.75)c \ar@{<-}[r]_b & \bullet \ar[u]_b \ar@{}@(r,d)[]_{v_2} }\end{xy}</m>
  690 B (108 words) - 15:06, 19 May 2017
• Cauchy-Schwarz inequality for inner product spaces
  Let {{M|\langle\cdot,\cdot\rangle:X\times X\rightarrow\mathbb{K} }} be an [[inner product]] so {{M|(X,\langle\cdot,\cdot\rangle)}} is an [[ **#* Consider {{M|\lambda\in\mathbb{K} }} then:
  6 KB (1,279 words) - 13:00, 4 April 2017
• Poisson distribution
  |data2={{MM|\mathbb{P}[X\eq k]:\eq e^{-\lambda}\frac{\lambda^k}{k!} }} |label3=[[Cumulative distribution function|c.d.f]]
  8 KB (1,401 words) - 00:52, 20 July 2018
• Binomial distribution
  |data1={{M|\mathbb{P}[X\eq k]:\eq{}^n\text{C}_k\ p^k(1-p)^{n-k} }} We are interested in getting exactly {{M|k}} {{M|y}}s. (specifically [[how many combinations]] there are of various {{
  4 KB (653 words) - 13:11, 22 September 2017
• Geometric distribution
  {{Stub page|grade=C|msg=Removed previous stub message and demoted [[User:Alec|Alec]] ([[User ta ...that the ''first'' success occurs on the {{M|k^\text{th} }} trial, for {{M|k\in\mathbb{N}_{\ge 1} }}.
  3 KB (557 words) - 15:14, 16 January 2018
• The sum of two random variables with Poisson distributions is a Poisson distribution itself
  {{Requires work|grade=C|msg=There's still some work to do on this page, but the gist is very much p We will show that {{M|\forall k\in\mathbb{N}_0\big[\P{Z\eq k}\eq\P{Z'\eq k}\big]}}
  3 KB (536 words) - 22:46, 4 November 2017
• Mdm of the Poisson distribution
  ...{MM|:\eq\sum^\infty_{k\eq 0}\big\vert X-\mathbb{E}[X]\big\vert\cdot\P{X\eq k} }} ...} {{MM|\eq e^{-\lambda}\ \sum^\infty_{k\eq 0}\frac{\lambda^k}{k!}\big\vert k-\lambda\big\vert}}
  7 KB (1,308 words) - 00:27, 8 November 2017
• Alec's taxonomy of units
  ...ent for each! Notice we can speak of "twice as hot" with {{M|{}^\circ\text{K} }} (kelvin) though. - Kelvin is a '''real''' measure, the next category. | colspan="3" | <center>'''Kelvin {{M|\mathbf{ {}^\circ\text{K} } }}'''</center>
  4 KB (592 words) - 07:29, 11 December 2017
• Poisson mixture
  * {{XXX|Link here}} {{C|If we have a Poisson distribution and each of its events being noticed i.i. Let {{M|\mathcal{C} }} be an {{M|N}}-valued [[random variable]] where {{M|N:\eq\{1,2,3,\ldots,
  3 KB (509 words) - 00:41, 20 July 2018