How to Show an Identity Map is Continuous

Modern General Topology

In North-Holland Mathematical Library, 1985

Proof

We consider the identity mapping f of X onto X. Then by Q) we can extend f to a uniformly continuous mapping g of $X^{\prime }$ into $\stackrel{\_}{X}$ . In the same way, we can extend f ⁻¹ to a uniformly continuous mapping h of $\stackrel{\_}{X}$ into $X^{\prime }$ . Suppose that $p\in \widetilde{X,}\text{undefined}q\in \stackrel{\_}{X}$ and g(p) = q. Let

$\mathcal{G}=\left\{V\left(q\right)\cap X|V\left(q\right)\text{ is a nbd of}q\text{ in}\widehat{X}\right\}.$

Putting

$ℱ=\left\{F|F\subset \tilde{X},F\supset f^{-1}\left(G\right)\text{ for some}G\in \mathcal{G}\right\},$

we get a Cauchy filter $ℱ$ of $X^{\prime }$ . As seen in the proof of Q),

On the other hand, we can assert that $ℱ\to p$ in $X^{\prime }$ . To see it, let U(p) be a given nbd of p in $X^{\prime }$ . Then by the definition of g,

Hence it follows from the definition of $\mathcal{G}$ that

$\begin{array}{ll}G\cap f\left(U\left(p\right)\cap X\right)\ne 0\hfill & \text{for every}G\in \mathcal{G},\hfill \end{array}$

which implies

$\begin{array}{ll}{f}^{-1}\left(G\right)\cap U\left(p\right)\ne 0\hfill & \text{for every}G\in \mathcal{G}\text{ in}\tilde{X}.\hfill \end{array}$

Thus $p\in \overline{f^{-1}\left(G\right)}$ , which means that p is a cluster point of $ℱ$ . Since $ℱ$ is a Cauchy filter, by N), $ℱ\to p.$ Thus from (1) we can conclude that h(q) = p, i.e. h = g ⁻¹. This proves that g is a unimorphic mapping of $X^{\prime }$ onto $\stackrel{\_}{X}$ , proving the proposition.

Combining O), P), Q) and R) we obtain the following theorem.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0924650909700558

Handbook of Computability Theory

Yuri L. Ershov , in Studies in Logic and the Foundations of Mathematics, 1999

Examples.

(1)

The identity mapping id _N : N → N is a numbering; denote by N the corresponding numbered set (N, id _N ).

(2)

If P _ω(N) is the set of all finite subsets of N, then a canonical numbering of P _ω(N) is a numbering γ: N → P _ω(N) defined by its inverse mapping γ⁻¹: P _ω(N) → N in the following way:

${\gamma }^{-1}(\varnothing )\leftrightharpoons 0;{\gamma }^{-1}\left(\left\{a_0<\mathrm{\dots }<a_k\right\}\right)\leftrightharpoons {\displaystyle \sum _{i⩽k}{2}^{ak}}.$

Denote by Γ the numbered set (P _ω(N), γ).

(3)

The mapping c: N ² → N defined by

$c\left(x,y\right)\leftrightharpoons \frac{{\left(x+y\right)}^2+3x+y}{2}$

is one-to-one and surjective. The inverse mapping c ⁻¹: N → N ² is a numbering of the set N ². Denote by r and l unary (recursive) functions such that c ⁻¹(x) = 〈l(x), r(x)〉 for x ∈ N.

Let S ₀ ⊆ S, v be a numbering of S, and v ₀ be a numbering of S ₀. The numbering v ₀ is reducible to v (v ₀ ⩽ v), if there exists an unary recursive function f such that v ₀ = vf. If v and v ₀ are numberings of S, v ₀ ⩽ v, and v ⩽ v ₀, then v ₀ and v are called equivalent (v ₀ ≡ v). If S ₀ ⊆ S, v ₀: N → S ₀, v: N → S are numberings and v ₀ ⩽ v, then we will use the notation S ₀ ⩽ S (for S ₀ = (S ₀, v ₀), S = (S, v)).

Denote by Nu(S) the set of all numberings of a set S. The relation ≡ on elements of Nu(S) is an equivalence. Denote by L(S) the factor set Nu(S)/_≡. The reducibility relation ⩽ induces an order on L(S) which will be denoted also by ⩽.

The ordered set 〈L(S), ⩽〉 is an upper semilattice as the following consideration simply:

Let v ₀, v ₁ ∈ Nu(S). Define a numbering v ₀ ⊕ v ₁ of the set S by

$\left({\upsilon }_0\oplus {\upsilon }_1\right)\left(2x\right)\leftrightharpoons {\upsilon }_0\left(x\right);\left({\upsilon }_0\oplus {\upsilon }_1\right)\left(2x+1\right)\leftrightharpoons {\upsilon }_1\left(x\right),x\in N.$

It is easy to verify that:

(1)

v ₀ ⩽ v ₀ ⊕ v ₁, v ₁ < v ₀ ⊕ v ₁;

(2)

if v ∈ Nu(S), v ₀ ⩽ v, v ₁ ⩽ v, then v ₀ ⊕ v ₁ ⩽ v.

These properties imply that if we denote by [v] the class of all numberings equivalent to v, then [v ₀ ⊕ v ₁] is the least upper bound of [v ₀] and [v ₁] in 〈L(S), ⩽〉.

A numbering v: N → S induces a numbering equivalence η_v on N:

${\eta }_{\nu }\leftrightharpoons \left\{〈x,y〉|x,y,\in N,\upsilon \left(x\right)=\upsilon \left(y\right)\right\}.$

A numbering v is called decidable (positive, negative), if η _v is a recursive (recursively enumerable, complement to recursively enumerable) subset of N ². A numbering v is called single-valued, if ${\eta }_{\nu }=\left\{〈x,y〉|x\in N\right\}.$

If v is a positive numbering of S, then [v] is a minimal element of 〈L(S), ⩽〉.

An upper semilattice 〈L(S), ⩽〉 has the least element iff S is finite.

If S is infinite there are continuum many minimal elements in 〈L(S), ⩽〉, namely those which contain single-valued numberings.

If S is finite and contains more than one element, or S is countably infinite, then there is no biggest element in 〈L(S), ⩽〉.

It is obvious if S ₀ and S ₁ are countably infinite, then upper semilattices 〈L(S ₀), ⩽〉 and 〈L(S ₁), ⩽〉 are isomorphic. (Every one-to-one mapping φ of S ₀ onto S ₁ induces bijection v ↦ φv, v ∈ Nu(S ₀), of the set Nu(S ₀) onto Nu(S ₁), which in turn induces an isomorphism of 〈L(S ₀), ⩽〉 and 〈L(S ₁), ⩽〉.)

Far less obvious is the following:

If S ₀ and S ₁ are finite and contain more than one element, then 〈L(S ₀), ⩽〉 and 〈L(S ₁), ⩽〉 are isomorphic.

This follows from the explicit description of the upper semilattice 〈L(S), ⩽〉 for a finite S (|S| ≥ 2) established in Ershov [1975] (see also Ershov [1977]). Note also that if |S| = 2, then 〈L(S), ⩽〉 is naturally isomorphic to the upper semilattice 〈L_m , ⩽〉 of m-degrees of proper subsets of N.

Sometimes it is convenient to assume that the empty set ∅ has some (unique) numbering 0 which is reducible to any numbering of any set. Denote by Nu*(S) the set of all numberings of all subsets (including ∅) of S. The reducibility relation ⩽ is a preorder on Nu*(S). Let L*(S) be the factor set Nu*(S)/_≡ and ⩽ be the order on L*(S) induced by the reducibility relation.

For v ₀, v ₁ ∈ Nu*(S) define

$v_0\oplus v_1=v_1\oplus v_0=v_{0}if{v}_1=0,$

$\left(v_0\oplus v_1\right)\left(2x\right)\leftrightharpoons v_0\left(x\right);\left(v_0\oplus v_1\right)\left(2x+1\right)\leftrightharpoons v_1\left(x\right),x\in N,$

if v ₀ ≠ 0, v ₁ ≠ 0. Note that if v ₀ is a numbering of S ₀ ⊆ S and v ₁ is a numbering of S ₁ ⊆ S, then v ₀ ⊕ v ₁ is a numbering of S ₀ ∪ S ₁ ⊆ S.

The element [v ₀ ⊕ v ₁] of the ordered set 〈L*(S), ⩽〉 is the least upper bound of the elements [v ₀] and [v ₁].

The upper semilattice 〈L*(S), ⩽〉 satisfies the following distributivity property: If [v] ⩽ [v ₀ ⊕ v ₁], then there exist v′ ₀, v′ ₁ ∈ Nu*(S) such that v′₀ ⩽ v ₀, v′ ₁ ⩽ v ₁, and

v ≡ v′ ₀ ⊕ v′ ₁.

For finite S the upper semilattice 〈L(S), ⩽〉 is also distributive.

One of the main methodological problems of the theory of numberings is to select the "proper" numbering for the sets in observation.

One of the general approaches is to select the principal numbering.

We demonstrate it initially in the following environment: let S = (S, v) be a numbered set and S ₀ ⊆ S. A numbering v ₀: N → S ₀ is called v-computable or computable relative to S, if v ₀ ⩽ v, i.e. v ₀ is reducible to v. A subset S ₀ ⊆ S is called v-computable subset of S, if there exists a v-computable numbering of S ₀. A v-computable numbering v ₀: N → S ₀ is called principal computable numbering relative to S, if for every v-computable numbering v′ ₀: N → S ₀ we have v′₀ ⩽ v ₀. If v ₀ is a principal numbering of S ₀, then we call S ₀ a principal subset of S, and S ₀ = (S₀, v ₀) the principal numbered subset of S.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0049237X99800305

Modern General Topology

In North-Holland Mathematical Library, 1985

Corollary 2

Let B(X) be a compact T₂-space in which X is a dense subset. Then there is a continuous mapping g of β(X) onto B(X) which keeps every point of X fixed and maps β(X) − X onto B(X) − X.

Proof

Let us denote by f the identity mapping which maps $X\subset \beta \left(X\right)$ onto $X\subset B\left(X\right)$ , namely f(p) = p for every $p\in X$ . By use of Corollary 1, we can extend f to a continuous mapping g of β(X) into B(X). Since g is continuous, g(β(X)) is compact and therefore closed in B(X). This combined with $\overline{X}=B\left(X\right)$ implies g(β(X)) = B(X). To prove $g\left(\beta \left(X\right)-X\right)\subset B\left(X\right)-X$ , we suppose that p is a given point of β(X) − X. Put

$ℱ=\left\{U\cap X|U\text{is}\text{a}\text{nbd}\text{of}p\text{}\text{in}\beta \left(X\right)\right\};$

then $\mathcal{F}$ is a filter basis in β(X) converging to p. Since g is continuous, by II.6.B),

Observe that $g\left(ℱ\right)=ℱ$ because each element of $\mathcal{F}$ is a subset of X on which g = f. Thus $ℱ\to g\left(p\right)$ in B(X). Since $\mathcal{F}$ converges to no point of X, we conclude that

which proves the assertion.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0924650909700534

Reliable Methods for Computer Simulation

In Studies in Mathematics and Its Applications, 2004

6.7.1 Residual based estimates

The simplest way is to set

(6.7.6) $y^{*}=y_1^{*},$

which means that II is the identity mapping of Y ^*to itself. In this case,

$D\left(\Lambda v,y_1^{*}\right)=0$

and ℳ_⊕(v, β,y ₁ ^*) contains only the second term, which upon minimization with respect to β gives the first form of the majorant

(6.7.7) $M_{\oplus }^{\left(1\right)}\left(v\right)=\frac{1}{2}|l+{\Lambda }^{*}A\Lambda v|,$

which implies the estimate

(6.7.8) $\left|\right||\Lambda \left(v-u\right)|\left|\right|\le |l+{\Lambda }^{*}A\Lambda v|=\underset{w\in V_0}{\sup }\frac{\left(g,w\right)+\left(A\Lambda v,\Lambda w\right)}{\left|\right||\Lambda w|\left|\right|}.$

If v is obtained by the finite element method (so that v coincides with a Galerkin approximation u _h ∈ V _h :=V _0h +u ₀), then the right-hand side of (6.7.8) is estimated by the method presented in Chapter 4. As a result, we obtain residual type a posteriori error estimates that involve integral terms associated with finite elements and interelement jumps.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0168202404800079

ALGORITHMIC INFORMATION THEORY

Peter D. Grünwald , Paul M.B. Vitányi , in Philosophy of Information, 2008

4.2 Coding Preliminaries

Strings and Natural Numbers Let χ be some finite or countable set. We use the notation χ* to denote the set of finite strings or sequences over χ. For example,

${\left\{\mathrm{0,1}\right\}}^{*}=\left\{\varepsilon ,\mathrm{0,1,00,01,10,11,000},\dots \right\},$

with ∈ denoting the empty word" with no letters. We identify the natural numbers N and {0, 1}* according to the correspondence

(7) $\left(0,\varepsilon \right),\left(\mathrm{1,0}\right),\left(\mathrm{2,1}\right),\left(\mathrm{3,00}\right),\left(\mathrm{4,01}\right),\dots$

The length l(x) of x is the number of bits in the binary string x. For example, l(010) = 3 and l(∈) = 0. If x is interpreted as an integer, we get l(x) = [log(x + l)] and, for x ≥ 2,

(8) $\left[logx\right]\le l\left(x\right)\le \left[logx\right].$

Here, as in the sequel, ┌x┐ is the smallest integer larger than or equal to x, └x┘ is the largest integer smaller than or equal to x and log denotes logarithm to base two. We shall typically be concerned with encoding finite-length binary strings by other finite-length binary strings. The emphasis is on binary strings only for convenience; observations in any alphabet can be so encoded in a way that is 'theory neutral'.

Codes We repeatedly consider the following scenario: a sender (say, A) wants to communicate or transmit some information to a receiver (say, B). The information to be transmitted is an element from some set χ. It will be communicated by sending a binary string, called the message. When B receives the message, he can decode it again and (hopefully) reconstruct the element of X that was sent. To achieve this, A and B need to agree on a code or description method before communicating. Intuitively, this is a binary relation between source words and associated code words. The relation is fully characterized by the decoding function. Such a decoding function D can be any function D: {0, 1}* → χ. The domain of D is the set of code words and the range of D is the set of source words. D(y) = x is interpreted as "y is a code word for the source word x". The set of all code words for source word x is the set D ⁻¹(x) = {y: D(y) = x}. Hence, E = D ⁻¹ can be called the encoding substitution (E is not necessarily a function). With each code D we can associate a length function L_D : χ → N such that, for each source word x, L_D (x) is the length of the shortest encoding of x:

$L_D\left(x\right)=min\left\{l\left(y\right):D\left(y\right)=x\right\}.$

We denote by x* the shortest y such that D(y) = x; if there is more than one such y, then x* is defined to be the first such y in lexicographical order.

In coding theory attention is often restricted to the case where the source word set is finite, say χ = {1, 2, …, N}. If there is a constant l ₀ such that l(y) = l ₀ for all code words y (equivalently, L(x) = l ₀ for all source words x), then we call D a fixed-length code. It is easy to see that l ₀ ≥ log N. For instance, in teletype transmissions the source has an alphabet of N = 32 letters, consisting of the 26 letters in the Latin alphabet plus 6 special characters. Hence, we need l ₀ = 5 binary digits per source letter. In electronic computers we often use the fixed-length ASCII code with l ₀ = 8.

Prefix-free code In general we cannot uniquely recover x and y from E(xy). Let E be the identity mapping. Then we have E(00) E(00) = 0000 = E(0) E(000). We now introduce prefix-free codes, which do not suffer from this defect. A binary string x is a proper prefix of a binary string y if we can write y = xz for z ≠ ∈. A set {x, y,…} ⊆ {0, 1}* is prefix-free if for any pair of distinct elements in the set neither is a proper prefix of the other. A function D: {0, 1}* → N defines a prefix-free code ³ if its domain is prefix-free. In order to decode a code sequence of a prefix-free code, we simply start at the beginning and decode one code word at a time. When we come to the end of a code word, we know it is the end, since no code word is the prefix of any other code word in a prefix-free code. Clearly, prefix-free codes are uniquely decodable: we can always unambiguously reconstruct an outcome from its encoding. Prefix codes are not the only codes with this property; there are uniquely decodable codes which are not prefix-free. In the next section, we will define Kolmogorov complexity in terms of prefix-free codes. One may wonder why we did not opt for general uniquely decodable codes. There is a good reason for this: It turns out that every uniquely decodable code can be replaced by a prefix-free code without changing the set of code-word lengths. This follows from a sophisticated version of the Kraft inequality [Cover and Thomas, 1991, Kraft-McMillan inequality, Theorem 5.5.1]; the basic Kraft inequality is found in [Harremoës and Topsøe, 2008], Equation 1.1. In Shannon's and Kolmogorov's theories, we are only interested in code word lengths of uniquely decodable codes rather than actual encodings. The Kraft-McMillan inequality shows that without loss of generality, we may restrict the set of codes we work with to prefix-free codes, which are much easier to handle.

Codes for the integers; Pairing Functions Suppose we encode each binary string x = x ₁ x ₂ … x_n as

The resulting code is prefix-free because we can determine where the code word $\overline{x}$ ends by reading it from left to right without backing up. Note $l\left(\overline{x}\right)=2n+1;$ thus, we have encoded strings in {0, 1}* in a prefix-free manner at the price of doubling their length. We can get a much more efficient code by applying the construction above to the length l(x) of x rather than x itself: define x' = l(x)x, where l(x) is interpreted as a binary string according to the correspondence (7). Then the code that maps x to x' is a prefix-free code satisfying, for all x ∈ {0, 1}*, l(x') = n + 2 log n + 1 (here we ignore the 'rounding error' in (8)). We call this code the standard prefix-free code for the natural numbers and use L N (x) as notation for the codelength of x under this code: L _N (x) = l(x'). When x is interpreted as a number (using the correspondence (7) and (8)), we see that L _N (x) = log x + 2 log log x + 1.

We are often interested in representing a pair of natural numbers (or binary strings) as a single natural number (binary string). To this end, we define the standard 1-1 pairing function 〈⋅,⋅〉: N × N → as 〈x, y〉 = x'y (in this definition x and y are interpreted as strings).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444517265500133

Euclidean Geometry

Barrett O'Neill , in Elementary Differential Geometry (Second Edition), 2006

3.4 Example

(1) Translations. All translations are orientation-preserving. Geometrically this is clear, and in fact the orthogonal part of a translation T is just the identity mapping I, so sgn T = det I = +1.

(2) Rotations. Consider the orthogonal transformation C given in Example 1.2, which rotates R ³ through angle θ around the z axis. Its matrix is

$\left(\begin{array}{ccc}\text{cos}\theta & -\text{sin}\theta & 0\\ \text{sin}\theta & \text{cos}\theta & 0\\ 0& 0& 1\end{array}\right).$

Hence sgn C = det C = +1, so C is orientation-preserving (see Exercise 4).

(3) Reflections. One can (literally) see reversal of orientation by using a mirror. Suppose the yz plane of R ³ is the mirror. If one looks toward that plane, the point p = (p ₁, p ₂, p ₃) appears to be located at the point

(Fig. 3.4). The mapping R so defined is called reflection in the yz plane. Evidently it is an orthogonal transformation, with matrix

FIG. 3.4.

Thus R is an orientation-reversing isometry, as confirmed by the experimental fact that the mirror image of a right hand is a left hand.

Both dot and cross product were originally defined in terms of Euclidean coordinates. We have seen that the dot product is given by the same formula,

$\text{v}\cdot \text{w}=(\sum v_i{\text{e}}_i)\cdot (\sum w_i{\text{e}}_i)=\sum v_i{\text{w}}_i,$

no matter what frame e ₁, e ₂, e ₃ is used to get coordinates for v and w. Almost the same result holds for cross products, but orientation is now involved.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780120887354500079

DIFFERENTIAL CALCULUS ON BANACH SPACES

YVONNE CHOQUET-BRUHAT , CÉCILE DEWITT-MORETTE , in Analysis, Manifolds and Physics, 2000

Answer 1d

If the diffeomorphism f_s : S ×R → S × R is

$\begin{array}{lll}\widehat{x}=x\hfill & \text{and}\hfill & \widehat{t}=t+s\hfill \end{array}$

the mapping f ^* _s is the identity mapping. Therefore, a lagrangian L(u, Du,.) τ is invariant by f_s if and only if in coordinates x ^α = (xⁱ , x ⁰) adapted to the product structure of S ×R, the lagrangian $\overline{L}\left(u^a,{\text{D}}_{\alpha }{u}^a,x^i,x^0\right)\rho \left(x^i,x^0\right)$ does not depend explicitly on x ⁰. The corresponding conserved current is

$\begin{array}{ll}{J}^{\alpha }=-\frac{\partial L}{\partial \left({\partial }_{\alpha }\cdot u^a\right)}{\partial }_0{u}^a+{\delta }_0^{a}L,\hfill & \alpha =0,1,\mathrm{\dots },d-1.\hfill \end{array}$

The conservation law is

${\text{D}}_{\alpha }{J}^{\alpha }\equiv \frac{1}{\rho }{\partial }_{\alpha }\left(J^{\alpha }\rho \right)=0$

which implies for any domain Ω of V with boundary ∂ Ω :

(3) $0={\displaystyle \underset{\Omega }{\int }{\text{D}}_{\alpha }{J}^{\alpha }\tau }={\displaystyle \underset{\partial \Omega }{\int }{J}^{\alpha }{\sigma }_{\alpha }},$

where σ _α is the (d − 1)-form

${\sigma }_{\alpha }=\rho \text{d}{x}^0\wedge \cdots \wedge \widehat{\text{d}{x}^{\alpha }}\wedge \cdots \wedge \text{d}{x}^{d-1},$

where dx ^α is suppressed. If we take Ω = S × [t ₀, t ₁] and S is compact we have ∂ Ω = S _{t 0} ∪ S _{t 1} and (3) reads

${\displaystyle \underset{S_{t_1}}{\int }{J}^0\rho \text{d}{x}^1\mathrm{\dots }\text{d}{x}^{d-1}={\displaystyle \underset{S_{t_1}}{\int }{J}^0\rho \text{d}{x}^1\mathrm{\dots }\text{d}{x}^{d-1}.}}$

If S is not compact, but J ^α has compact support we have the same formula. If J ^α is a limit of functions with compact support we may have the same formula. Otherwise supplementary terms can appear.

Answer 1e: In local coordinates, x ⁰ ∈ R, xⁱ , with i = 1, 2, …, d − 1, coordinates on S,

$L=g^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u+m^2{u}^2.$

L does not depend explicitly on x ⁰ (since ∂ ₀ g ^{α β} = 0), and the conserved current is

$\begin{array}{l}{J}^0=L-2g^{\alpha 0}{\partial }_{\alpha }u{\partial }_{0}u=-g^{00}{\left({\partial }_{0}u\right)}^2+g^{ij}{\partial }_{i}u{\partial }_{j}u+m^2{u}^2,\\ J^i=-2g^{\alpha i}{\partial }_{\alpha }u{\partial }_{0}u.\end{array}$

The energy density J ⁰ on S_t is positive.

Note: Compare the current obtained here with the relations on pp. 513-514.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978044450473950004X

Mathematical preliminaries

Peter R. Massopust , in Fractal Functions, Fractal Surfaces, and Wavelets (Second Edition), 2016

4.2 Hölder spaces

Throughout this section, Ω denotes a nonempty open subset of ${\mathbb{R}}^n$ , n ≥ 1. Let $k\in {\mathbb{N}}_0$ and 0 < α ≤ 1. To fill in the gaps between the discrete ladder of smoothness spaces $C_{\mathbb{R}}^k(\mathrm{\Omega }):=C^k(\mathrm{\Omega },\mathbb{R})$ , $k\in {\mathbb{N}}_0$ , we introduce the so-called (inhomogeneous) Hölder spaces.

Notation

For notational simplicity, we drop the subscript $\mathbb{R}$ from all function spaces whose elements map to $\mathbb{R}$ .

The Hölder spaceC ^{k, α} (Ω) is defined as the linear space of all k-fold continuously differentiable bounded functions $f:\mathrm{\Omega }\to \mathbb{R}$ such that there exists a constant c > 0 with

$\frac{|(D^{\ell }f)(x)-(D^{\ell }f)(y)|}{\parallel x-y\parallel {}^{\alpha }}\le c<\infty ,\forall x,y\in \mathrm{\Omega }.$

Here, we used multiindex notation, writing

$D^{\ell }:=\frac{{\partial }^{{\ell }_1+{\ell }_2+\cdots +{\ell }_n}}{\partial x_1^{{\ell }_1}\partial x_2^{{\ell }_2}\cdots \partial x_n^{{\ell }_n}}$

for the differential monomial D^ℓ of order |ℓ| := ℓ ₁ + ℓ ₂ + ⋯ + ℓ _n . As usual, D ⁰ := I, where I denotes the identity operator on ${\mathbb{R}}^n$ .

It can be shown that the linear spaces C ^{k, α} (Ω) when endowed with the norms

$\parallel f\parallel {}_{k,\alpha }:=\underset{\genfrac{}{}{0ex}{}{|\ell |\le k}{x\in {\mathbb{R}}^n}}{sup}\{|(D^{\ell }f)(x)|\}+\underset{\genfrac{}{}{0ex}{}{|\ell |=k}{\genfrac{}{}{0ex}{}{x,y\in {\mathbb{R}}^n}{x\ne y}}}{sup}\left\{\frac{|(D^{\ell }f)(x)-(D^{\ell }f)(y)|}{\parallel x-y\parallel {}^{\alpha }}\right\}$

become Banach spaces. For completeness we also define

$C^{0,0}(\mathrm{\Omega }):=C(\mathrm{\Omega }),C^{k,0}(\mathrm{\Omega }):=C^k(\mathrm{\Omega }).$

Definition 46

Suppose X and Y are normed spaces. An embedding of X into Y, written X↪Y, is such that:

1.

X is a subvector space of Y;

2.

the identity mapping $X\ni x\stackrel{j}{\mapsto }j(x):=x\in Y$ is continuous; that is,

$\exists c>0\forall x\in X:\parallel j(x)\parallel {}_Y\le c\parallel x\parallel {}_X.$

The mapping j is called an embedding (of X into Y).

Using the definition of the respective norms, we can easily show the validity of the following embeddings: Let 0 < α < β ≤ 1. Then

1.

$C^{k+1}(\overline{\mathrm{\Omega }})\hookrightarrow C^k(\overline{\mathrm{\Omega }})$ ;

2.

$C^{k,\alpha }(\overline{\mathrm{\Omega }})\hookrightarrow C^k(\overline{\mathrm{\Omega }});$

3.

$C^{k,\beta }(\overline{\mathrm{\Omega }})\hookrightarrow C^{k,\alpha }(\overline{\mathrm{\Omega }})$ ,

where the spaces $C^k(\overline{\mathrm{\Omega }}$ and $C^{k,\alpha }(\overline{\mathrm{\Omega }})$ are understood as subspaces of C ^k (Ω) and C ^{k, α} (Ω), respectively, consisting of all functions in C ^k (Ω) or C ^{k, α} (Ω), which together with all their derivatives are continuous on the boundary ∂Ω of Ω.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128044087000011

Handbook of Algebra

V.K. Kharchenko , in Handbook of Algebra, 1996

3.3 Canonical sheaf

In this paragraph we shall present the closure $\widehat{RE}$ in the topology described above as a ring of global sections of a certain sheaf over the structure space of the extended centroid. Such a presentation is useful since it enables one to see clearly the function-theoretic intuition employed when studying a semiprime ring. It is worthwhile adding that in the case of a prime ring all these constructions degenerate and their essence and meaning is to reduce the process of studying semiprime rings to that of studying prime rings (i.e. stalks of a canonical sheaf).

Roughly speaking, the ring of global sections of a sheaf is a set of all continuous functions on a topological space, the only difference being that at every point these functions assume values in their own (local) rings. Let us give the exact definitions.

3.3.1 Definition

Let there be given a topological space X, and for any open set U let there be given a ring (group or, more generally, an object of a certain fixed category) ℛ(U), and let for any two open sets U ⊂ V there be given a homomorphism ${\rho }_U^V:\Re (V)\to \Re (U)$ .

This system is called a presheaf of rings, provided the following conditions are met:

(1).

if U is empty, then ℛ(U) is the zero ring;

(2).

${\rho }_U^U$ is the identity mapping;

(3).

for any open sets U ⊆ V ⊆ W we have ${\rho }_U^W={\rho }_U^V{\rho }_V^W$ .

Such a presheaf will be denoted by one letter, ℛ.

The simplest example of a presheaf is the presheaf of all functions on X with the values in a ring A. In this case ℛ(U) consists of all functions on U with values in A, and for U ⊂ V the homomorphism ${\rho }_U^V$ is the restriction of the function determined on V to the subset U.

In order to extend the intuition of this example to the case of any presheaf, the homomorphisms ${\rho }_U^V$ are called restriction homomorphisms. The elements of (the ring) ℛ(U) are called the sections of the presheaf ℛ over U. Sections of ℛ over X are called global. Thus, ℛ(U) is a ring of sections of the presheaf ℛ over U; ℛ(X) is a ring of global sections.

Returning to the example of the presheaf of all functions on X, let us assume the topological space X to be the union of open sets U_α . Then any function on X is uniquely determined by its restrictions to the sets U_α . Moreover, if on every U_α a function f_α is given, such that the restrictions of f _α and f_β coincide on U_α ∩ U_β , then there exists a function on X, such that every f_α is its restriction to U_α .

These properties can be formulated for any presheaf and they single out an extremely important class of presheaves.

3.3.2 Definition

A presheaf R on a topological space X is called a sheaf if for each open set U ⊂ X and every open cover U = ∪ U_α the following conditions are met:

(1).

if ${\rho }_{U_{\alpha }}^U(s_1)={\rho }_{U_{\alpha }}^U(s_2)$ for s ₁, s ₂ ∈ ℛ(U) and all α, then s ₁ = s₂;

(2).

if s_α ∈ ℛ(U_α ) are such that for any α, β the restrictions of s_α and s_β on U_α∩U_β coincide, then there exists an element s ∈ ℛ(U), whose restriction to U_α is equal to s_α for all α.

Let us try to consider an arbitrary sheaf as a sheaf of functions on the space X. To this end it is necessary to determine the value of s(x) for the section s ∈ ℛ(U) at any point x ∈ U. We have the elements $s(V)={\rho }_V^U(s)$ for all open neighborhoods V of the point x which are contained in U. Thus it is natural to consider their 'limit', and so we have to introduce the direct limit

${\Re }_x=\underset{\overrightarrow{x\in V}}{\lim }\Re (V)$

with respect to the system of homomorphisms

$p_V^U:\Re \left(U\right)\to \Re \left(V\right).$

This limit is called the stalk of the sheaf (or a presheaf) ℛ at the point x.

Thus, a stalk element at the point x is determined by any section over an open neighborhood of x. And two sections, u, v ∈ ℛ(U), define the same element of the stalk at x if their restrictions to some open neighborhood of x coincide.

For any open set U ϶ x this gives a natural homomorphism ${\rho }_x^U:\Re (U)\to {\Re }_x$ , which maps a section to the stalk element determined by it. Thus we can define the value of a section s at the point x as ${\rho }_x^U(s)$ .

3.3.3 The construction

Let us now go over to constructing the canonical sheaf. Let C be a generalized centroid of a semiprime ring R. Let us denote by X the set of all its prime ideals but C. This set is called a spectrum of the ring C and is often denoted by Spec C. The elements of the spectrum are called points of the spectrum or simply points.

In order to put a topology on X, it is necessary to define the operation of closure. For a set A ⊂ X let us define the closure $\overline{A}$ as a set of all points containing the intersection

${\displaystyle \underset{p\in A}{\cap }p.}$

The topology obtained in this way is called the spectral topology.

In order to construct the sheaf over X, it is useful to know the structure of open sets in X (they are also called domains).

If e ∈ E is a central idempotent, then by U(e) we denote a set of all points p, such that e ∉ p.

Allowing for the fact that the product e (1 − e) = 0 belongs to any prime ideal, we see any point of the spectrum contains either e or 1−e, but not both simultaneously. Therefore,

(15) $\begin{array}{cc}U\left(e\right)\cup U\left(1-e\right)=X,& U\left(e\right)\cap U\left(1-e\right)=\varnothing .\end{array}$

On the other hand, the closure of U(e) contains only points q containing the intersection

${\displaystyle \underset{e\notin p}{\cap }p}={\displaystyle \underset{1-e\in p}{\cap }p.}$

The latter intersection contains the element 1 − e and, hence, 1 − e ∈ q, which implies e ∉ q, i.e. by definition, q ∈ U(e).

Now relations (15) show U.(e) to be an open and closed set simultaneously. Such a set is called a clopen set.

3.3.4 Lemma

Each clopen subset of X has the form U(e) for suitable idempotent e ∈ E.

3.3.5 Lemma

The sets U(e), e ∈ E form a fundamental system of open neighborhoods of X, i.e. any open set is presented as a union of sets of the type U(e).

Lemma 3.3.4 shows the set of central idempotents to be in a one-to-one correspondence with the set of closed domains of X. One can easily prove that this correspondence preserves the lattice operations and order relation:

(16) $\begin{array}{l}\begin{array}{cc}U\left(e_1{e}_2\right)=U\left(e_1\right)\cap U\left(e_2\right);& U\left(e_1+e_2-e_1{e}_2\right)\end{array}=U\left(e_1\right)\cup U\left(e_2\right),\\ U\left(e_1\right)\subseteq U\left(e_2\right)\iff e_1⩽e_2.\end{array}$

This circumstance allows one in a number of cases to identify central idempotents with closed domains and consider idempotents as objects consisting of points, which makes many considerations extremely vivid. The correspondence under discussion also preserves the exact upper bounds

$U\left(\underset{\alpha }{\sup }\left\{e_{\alpha }\right\}\right)=\overline{{\displaystyle \underset{\alpha }{\cup }U\left(e_{\alpha }\right),}}$

where on the right we have the closure of the union of the domains U(e_α).

3.3.6 Theorem

The space X = Spec C is an extremely disconnected, compact and Hausdorff topological space.

Recall that a topological space is called extremely disconnected if the closure of each open set is open (and of course closed).

3.3.7 Definition

Now we are completely ready to determine the canonical sheaf Γ = Γ(R). Let U be an open set. By formula (15), its closure Ū is open and has the form U(e) for some central idempotent. Let us set

$\Gamma \left(U\right)\stackrel{df}{=}\Gamma \left(\overline{U}\right)\stackrel{df}{=}\widehat{eRE}.$

Since the inclusion W ⊆ U implies $\overline{W}\subseteq \overline{U}$ , and this inclusion by formula (16) gives the inequality f ≤ e for the corresponding idempotents $U(f)=\overline{W}$ , U(e) = Ū, then the homomorphism x → xf acting from $e\widehat{RE}$ onto $f\widehat{RE}$ is defined, which will be viewed as the restriction homomorphism ${\rho }_W^U$ .

The validity of axioms (1)–(3) of a presheaf for Γ is absolutely obvious.

3.3.8 Theorem

The presheaf Γ determined above is a sheaf.

So, we have achieved our goal: the ring $\widehat{RE}$ is presented as the ring of global sections of a sheaf Γ. This sheaf is called the canonical sheaf of the ring R.

This sheaf satisfies another important condition: any section can be extended to a global one (sheaves satisfying this condition are called flabby) and, moreover, the restriction homomorphisms are retractions, so that the ring of global sections naturally contains all the rings of local sections $e\widehat{RE}\subseteq \widehat{RE}$ .

3.3.9 Definition

The support of a global section s, or more generally, the support of a subset $S\subseteq \widehat{RE}$ of global sections is defined as the difference 1 - f, where f is the largest central idempotent with Sf = 0.

From the function point of view the support of s is a set of all points where the function s has nonzero values.

This definition allows one to describe the stalks of the canonical sheaf.

3.3.10 Theorem

A section s belongs to the kernel of the natural homomorphism ρ_p iff the support of the element s belongs to p or, which is equivalent, $s\in p\widehat{RE}$ . The stalk Γ_p is a prime ring, isomorphic to the factor-ring

$\widehat{RE}/\widehat{RE}\cap p\widehat{RE}.$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S157079549680026X

Lie algebras of Lie groups, Kac-Moody groups, supergroups, and some specialized topics in finite- and infinite-dimensional Lie algebras

N. Sthanumoorthy , in Introduction to Finite and Infinite Dimensional Lie (Super)algebras, 2016

Complex structure on a real vector space and a Lie algebra over R [42]

Definition 263

Let V be a vector space of finite dimension over $\mathbb{R}.$ A complex structure on V is a $\mathbb{R}$ - linear endomorphism J of V such that J ² = −I, where $I:V\to V$ is the identity mapping. The above vector space V over $\mathbb{R}$ with a complex structure, J will become a vector space $\tilde{V}$ over $\mathbb{C}$ by taking,

$\begin{array}{l}\hfill (a+ib)x=ax+bJx,\end{array}$

for $x\in V;a,b\in \mathbb{R}.$ J ² = −I implies α(βx) = (αβ)x for $\alpha \in \mathbb{C}\text{and}x\in V.$ Moreover ${dim}_c\tilde{V}=\frac{1}{2}{dim}_{R}V$ and V is even dimension. Here $\tilde{V}$ becomes a complex vector space associated to V. If V is a vector space over $\mathbb{C},$ we can consider V as a vector space $V^{\mathbb{R}}$ over $\mathbb{R}.$ The multiplication by i on V gives a complex structure J on $V^{\mathbb{R}}$ and $V=V^{\mathbb{R}}.$

Definition 264

A Lie algebra $\mathcal{G}$ over $\mathbb{R}$ is said to have a complex structure J if J is a complex structure on the vector space $\mathcal{G}$ satisfying,

$\begin{array}{l}\hfill [X,JY]=J[X,Y]\text{for all}X,Y\in \mathcal{G}.\end{array}$

Hence (adX) ∘ J = J ∘ ad X for all X ∈ V. So, ad(JX) = J(adX), for all X ∈ V. Also it is clear that

$\begin{array}{l}\hfill [JX,JY]=-[X,Y].\end{array}$

Now the corresponding complex vector space $\tilde{\mathcal{G}}$ becomes a Lie algebra over C with the bracket operation is that of $\mathcal{G}.$ We have

$\begin{array}{ll}\hfill [(a+ib)X,(c+id)Y]& =[aX+bJX,cY+dJX]\hfill \\ \hfill & =ac[X,Y]+bcJ[X,Y]+adJ[X,Y]-bd[X,Y]\hfill \\ \hfill \Rightarrow [(a+ib)X,(c+id)Y]& =(a+ib)(c+id)[X,Y].\hfill \end{array}$

If $\mathcal{G}$ is a Lie algebra over C, then the vector space ${\mathcal{G}}^R$ has a complex structure J given by the multiplication i on $\mathcal{G}.$ Now ${\mathcal{G}}^R$ becomes a Lie algebra over R with the complex structure J. If ${\mathcal{G}}_0$ is a Lie algebra over R, then the complex vector space is $\mathcal{G}={({\mathcal{G}}_0)}^C=\{x+iy|x,y\in {\mathcal{G}}_0\}.$ Now the bracket operation on $\mathcal{G}$ is given by

$\begin{array}{l}\hfill [x_1+i{y}_1,x_2+i{y}_2]=[x_1,x_2]-[y_1,y_2]+i([y_1,x_2]+[x_1,y_2]).\end{array}$

This bracket is a bilinear operation over C. We have $\mathcal{G}={({\mathcal{G}}_0)}^C,$ called the complexification of the Lie algebra ${\mathcal{G}}_0.$ Moreover, ${\mathcal{G}}^R$ is a Lie algebra over R with complex structure J derived from the multiplication i on $\mathcal{G}.$ The following result can be easily established.

Proposition 61

[42] Let K ₀,K, and K ^R denote the Killing forms of the Lie algebras ${\mathcal{G}}_0$ , $\mathcal{G}$ , and ${\mathcal{G}}^R$ , respectively. Then

$\begin{array}{ll}\hfill K_0(X,Y)& =K(X,Y),\text{for}X,Y\in {\mathcal{G}}_0\text{and}\hfill \\ \hfill K^R(X,Y)& =2\text{Real part of}(K(X,Y)),\text{for}X,Y\in {\mathcal{G}}^R.\hfill \end{array}$

Definition 265

Let G be a Lie group of even dimension 2n with Lie algebra $\mathcal{G}$ . Assume that there exists a real linear map $J:\mathcal{G}\to \mathcal{G}$ such that

(i)

J ² = −I,

(ii)

[JX,Y ] = J[X,Y ].

Then J is a complex structure on the Lie algebra $\mathcal{G}$ . Now we can regard $\mathcal{G}$ as a complex Lie algebra with the multiplication by "i" to be the mapping J. Hence the map J gives $\mathcal{G}$ a complex vector space structure and the bracket is complex bilinear.

Similarly, if G is a complex Lie group, then we can define a map J on its Lie algebra $\mathcal{G}$ so that $\mathcal{G}$ becomes a complex Lie algebra and hence the exponential mapping is holomorphic from $\mathcal{G}$ into G. Conversely, if G is an even dimensional Lie group, we can find a map $J:\mathcal{G}\to \mathcal{G}$ so that G becomes a complex Lie group.

Example 88

Consider $GL(n,\mathbb{C}).$ Let G be a subgroup of $GL(n,\mathbb{C}).$ The Lie algebra $\mathcal{G}$ of G is a complex subalgebra of $gl(n,\mathbb{C})$ and $\mathcal{G}$ has complex structure.

Definition 266

[42] A Real form of a Lie algebra $\mathcal{G}$ over $\mathbb{C}$ is a sub algebra ${\mathcal{G}}_0$ of the real Lie algebra ${\mathcal{G}}^R$ such that ${\mathcal{G}}^R={\mathcal{G}}_0+J{\mathcal{G}}_0.$ Now any element $z\in \mathcal{G}$ can be written uniquely as z = x + iy where $x,y\in {\mathcal{G}}_0.$ So $\mathcal{G}$ is isomorphic to a complexification of ${\mathcal{G}}_0.$ The mapping $\sigma :x+iy\to x-iy$ is called the conjugation of $\mathcal{G}$ with respect to ${\mathcal{G}}_0.$ The mapping σ satisfies the following properties,

$\begin{array}{ll}\hfill \sigma (\sigma (X))& =X,\sigma (X+Y)=\sigma (X)+\sigma (Y)\hfill \\ \hfill \sigma (\alpha X)& =\overline{\alpha }\sigma (X),\sigma [X,Y]=[\sigma X,\sigma Y]\hfill \end{array}$

for $X,Y\in \mathcal{G},\alpha \in C.$

We state the following theorem proved in Helgason [42].

Theorem 75

Every semisimple Lie algebra $\mathcal{G}$ over C has a real form which is compact.

Let G be a connected Lie group with Lie algebra ${\mathcal{G}}^R.$ Let U be an analytic subgroup of G whose Lie algebra is the compact real form of $\mathcal{G}$ . Hence U is a maximal compact subgroup of $\mathcal{G}.$ Denote by $Ū,$ the universal covering group of U with $Z(Ū),$ the center. In Table 6.2, $Z(Ū)$ denotes center of $Ū$ with Z _p , a cyclic group of order p.

Table 6.2. The simple Lie algebras $\mathcal{G}$ over C, their Lie groups G and the corresponding compact real forms U (real forms which are compact) and their dimensions [42]

$\mathcal{G}$	G	U	dim U	Z $(\stackrel{-}{\text{U}})$
A n (n ≥ 1)	SL(n + 1,C)	SU(n + 1)	n(n + 2)	Z n+1
B n (n ≥ 2)	SO(2n + 1,C)	SO(2n + 1)	n(2n + 1)	Z 2
C n (n ≥ 3)	Sp(n,C)	Sp(n)	n(2n + 1)	Z 2
D n (n ≥ 4)	SO(2n,C)	SO(2n)	n(2n − 1)	Z 4 if n = odd
				Z 2 + Z 2 if n = even
E 6	$E_6^C$	E 6	78	Z 3
E 7	$E_7^C$	E 7	133	Z 2
E 8	$E_8^C$	E 8	248	Z 1
F 4	$F_4^C$	F 4	52	Z 1
G 2	$G_2^C$	G 2	14	Z 1

Here A _n ,B _n ,C _n ,D _n are classical structures and E ₆,E ₇,E ₈,F ₄,G ₂ are exceptional structures.

Definition 267

Let G be a closed linear group. For such a group, we get $C^{\infty }$ curves $c(t),t\in \mathbb{R}$ with the property that c(0) is the identity 1. Now we define, $\mathcal{G}=\{c^{\prime }(0)|c:\mathbb{R}\to G\mathrm{with}c(0)=1\text{and}c$ is smooth function from R to the group of matrices}. This $\mathcal{G}$ defined above is closed under scalar multiplication. Moreover, G is closed under group conjugation x↦gxg ⁻¹. Again for $x\in \mathcal{G}$ and c(t), a smooth curve in G with c(0) = 1, the map $R\to G$ sending t↦Ad(c(t))x is a smooth function in $\mathcal{G}$ .

Here "Ad" denotes adjoint representation of R in G corresponding to adjoint representation "ad" of R in $\mathcal{G},$ the Lie algebra of G.

Definition 268

Let X be an n × n real or complex matrix. We define the exponential of X (exp X) by, ${\mathrm{e}}^X={\sum }_{m=0}^{\infty }\frac{X^m}{m!}.$ It can be directly proved that, for any real and complex matrix X, the series $expX$ is convergent and it is a continuous function of X. Let $\mathcal{G}$ be a Lie algebra. For each $X\in \mathcal{G},$ there exists a unique analytic homomorphism θ of R into G such that $\stackrel{.}{\theta }(0)=X.$ Let $expX=\theta (1)$ such that $\stackrel{.}{\theta }(0)=X.$ The mapping $X\to expX$ of $\mathcal{G}$ into G ( $exp:gl(n,\mathbb{C})\to GL(n,\mathbb{C})$ ), defined by

$\begin{array}{l}\hfill expX={\mathrm{e}}^X=I+X+\frac{1}{2}{X}^2+\frac{1}{6}{X}^3+\cdots ,\end{array}$

is called the exponential mapping. It can be directly verified that $exp(t+s)X=exptX\cdot expsX$ for $s,t\in \mathbb{R}$ and $X\in \mathcal{G}.$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128046753000066

curnowphim1978.blogspot.com

Source: https://www.sciencedirect.com/topics/mathematics/identity-mapping

Share this post