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• Products
  The idea of the product may be traced to the image of a stack, which is a simple arrangement of mul [[image:product as a stack.png|center]]
  44 KB (7,951 words) - 02:21, 30 November 2015
• Functions in higher dimensions
  Why do we need to study multidimensional spaces? *abstract spaces designed to accommodate multiple ''variables'' (such as the stock and the c
  113 KB (19,680 words) - 00:08, 23 February 2019
• Forms in Euclidean spaces
  ...real line is, by definition, a collection $\phi$ of linear maps on tangent spaces: Furthermore, we glue the tangent spaces together to form the bundle: the direction from $n$ to $n+1$ is the opposit
  44 KB (7,778 words) - 23:32, 24 April 2015
• Inner product spaces: part 1
  ==Distances and angles in vector spaces== '''Question:''' What about other vector spaces? Especially infinite dimensional ones such as ${\bf F}([a,b])$.
  14 KB (2,404 words) - 15:04, 13 October 2011
• The algebra of oriented cells
  ...is:</TD> <TD>a finite field</TD> <TD>an infinite cyclic group</TD> <TD>an infinite field</TD> </TR> ...icient Theorem computes ''all'' others from $H(K;{\bf Z})$!</TD> <TD>other infinite fields: ${\bf C},{\bf Q}$</TD> </TR>
  36 KB (6,395 words) - 14:09, 1 December 2015
• The gradient
  ...{\bf R}^n$. Let's recast this expression, as before, in terms of the ''dot product'' with the increment of the independent variable: We used the ''Linearity of the dot product''. The “Linear Constant Multiple Rule” relies on the same property:
  74 KB (13,039 words) - 14:05, 24 November 2018
• Metric complexes
  ...develop the mathematics. There are two main exceptions. One needs the dot product and the norm for the following related concepts: In linear algebra, we learn how an inner product adds geometry to a vector space. We choose a more general setting.
  35 KB (5,871 words) - 22:43, 7 April 2016
• Continuous functions
  '''Theorem.''' Given two topological spaces $X,Y$, a function $f:X\to Y$ is continuous<!--\index{continuous function}-- '''Proposition.''' In case of metric spaces, a function $f:X\to Y$ is continuous if and only if ''the function commutes
  42 KB (7,138 words) - 19:08, 28 November 2015
• Calculus on cubical complexes
  ...R}^n$. We choose for now to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple ...at the discrete differential forms, as cochains, are organized into vector spaces, one for each degree. Let's review this first.
  36 KB (6,218 words) - 16:26, 30 November 2015
• Differential calculus of parametric curves
  *the dot product (the outcome is a scalar function!). ...product, $X=c(t) \cdot F(t)$, and the limit of the product is equal to the product of the limits:
  130 KB (22,842 words) - 13:52, 24 November 2018
• Discrete forms
  ...bf R}^n$. We choose here to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple ...at the discrete differential forms, as cochains, are organized into vector spaces, one for each degree. Let's review this first.
  35 KB (6,055 words) - 13:23, 24 August 2015
• Topology Illustrated -- Index
  *[[infinite dimensional space|infinite dimensional space]] *[[inner product|inner product]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Product topology
  [[image:convergence on the plane product.png|center]] Following this lead, for any two sets $X$ and $Y$ their [[product set]] is defined as the set of ordered pairs taken from $X$ and $Y$:
  8 KB (1,339 words) - 16:53, 27 August 2015
• Cubical calculus
  ...R}^n$. We choose for now to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into cubes, ${\mat ...ns<!--\index{differential form}-->, as cochains, are organized into vector spaces, one for each degree/dimension. Let's review this first.
  25 KB (4,238 words) - 02:30, 6 April 2016
• Compact spaces
  ...quate tool for studying topological spaces in general, we will look at all infinite subsets. ...hat this means for the open cover $\alpha$. We just discovered that this ''infinite'' cover
  19 KB (3,207 words) - 13:06, 29 November 2015
• Cochain complexes and cohomology
  However, as vector spaces they aren't the simplest ones, because they are infinite dimensional: Just as with continuous forms we define two new vector spaces:
  17 KB (2,592 words) - 14:38, 14 April 2013
• Linear algebra: course
  This is a one-semester course in linear algebra and vector spaces. An emphasis is made on the coordinate free analysis. The course mimics in *[[Vector spaces: introduction]]
  2 KB (279 words) - 15:05, 12 December 2012
• Cohomology
  Therefore, by the Classification Theorem of Vector Spaces, we have the following: ...omorphic to its dual'', the behaviors of the linear operators on these two spaces aren't “aligned”, as we will show. Moreover, the isomorphism is depende
  45 KB (6,860 words) - 16:46, 30 November 2015
• Topological spaces
  As an example, all “open”, as we have known them, intervals, finite or infinite, are open in ${\bf R}$: For example all “closed” intervals, finite or infinite, are closed in ${\bf R}$:
  27 KB (4,693 words) - 02:35, 20 June 2019
• Introduction to Topology by Gamelin and Greene
  ONE METRIC SPACES 4 Products of [[metric spaces]]
  2 KB (170 words) - 21:51, 4 May 2011
• Integer-valued calculus
  ...ings simpler. First, there are no limits. Second, the infinite-dimensional spaces of functions and forms are replaced with finite-dimensional ones. Third, th Invoking linear algebra now, we have Euclidean spaces with these generators and linear operators with these matrices:
  27 KB (3,824 words) - 19:07, 26 January 2019
• Duality: forms as cochains
  '''Exercise.''' Prove that, if the spaces are finite-dimensional, we have Therefore, by the ''Classification Theorem of Vector Spaces'', we have the following:
  29 KB (4,540 words) - 13:42, 14 March 2016
• Advanced Topology: exercises
  #Prove that $\Omega ^k({\bf R})$ is infinite dimensional. ...f discrete differential forms for the complex: [[Image:Describe the vector spaces of discrete differential forms for the complex below.png|center]]
  9 KB (1,487 words) - 18:18, 9 May 2013
• Vector spaces: introduction
  This is called ''[[scalar multiplication]]'', also known as "scalar product": Specific vector spaces:
  14 KB (2,238 words) - 17:38, 5 September 2011
• Linear Algebra by Messer
  ...s here are too terse in my opinion. I was good to see infinite dimensional spaces as an optional topic. Many good examples are given. In chapter 4, [[inner product spaces]] are introduced axiomatically. I'd prefer to put this topic at the end of
  2 KB (325 words) - 19:45, 7 March 2016
• Algebra of differential forms
  What about [[symmetry]]? That is, is it similar to the [[dot product]]: ...symmetric but [[anti-symmetric]]. That is, they are more like the [[cross product]]:
  11 KB (1,947 words) - 18:14, 22 August 2015
• New topological spaces from old
  ...en the [[product set]] $X \times Y$ becomes a topological space with the [[product topology]].</center> Infinite products are also possible.
  1 KB (199 words) - 20:20, 21 July 2011
• Lefschetz numbers in control theory
  Typically, the space of pairs of states and inputs is the product of the two: $N=M\times U.$ However nontrivial bundles are also common. For ...of $M,N,$ and $f$ is also continuous. This means that we have to consider spaces [[homeomorphic]] (or [[homotopy equivalent]]) to $M,N$ and maps [[homotopic
  17 KB (3,052 words) - 22:12, 15 July 2014
• Closed and exact forms
  Recall from [[linear algebra]], that each linear operator produces two vector spaces: ...nd use [[linearity]] later), where $dX$ is a basis $k$-form (i.e., [[wedge product]] and $A=A(x^1,...,x^n)$ is a coefficient function, twice continuously diff
  9 KB (1,423 words) - 20:53, 13 March 2013
• Cubical complex
  ...ly, a cubical cell<!--\index{cells}--> in the $N$-dimensional space is the product of vertices and edges: ...epresentation of the cube $P$. For each edge $A_i = [n_i, n_i + 1]$ in the product, we define a pair of ''opposite faces'' of $P$:
  29 KB (4,800 words) - 13:41, 1 December 2015
• Chain complexes
  *with $R={\bf R}$ (or other fields), the chain groups are vector spaces, and now <center> ''modules are vector spaces over rings''.</center>
  33 KB (5,293 words) - 03:06, 31 March 2016
• Calculus Illustrated
  8 The areas of infinite regions: improper integrals 4 Infinite series
  16 KB (1,933 words) - 19:50, 28 June 2021