book_name stringclasses 89
values | def_number stringlengths 12 19 | text stringlengths 5.47k 10k |
|---|---|---|
1075_(GTM233)Topics in Banach Space Theory | Definition 3.1.1 |
Definition 3.1.1. A basis \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) of a Banach space \( X \) is unconditional if for each \( x \in X \) the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}^{ * }\left( x\right) {u}_{n} \) converges unconditionally.
Obviously, \( {\left( {u}_{n}\right) }_{n = 1}^{\in... |
1129_(GTM35)Several Complex Variables and Banach Algebras | Definition 11.5 |
Definition 11.5. Let \( \left( {A, X,\Omega, p}\right) \) be a maximum modulus algebra. Let \( E \) be a subset of \( \Omega \) . For \( n \) an integer \( \geq 1 \), we say that \( \left( {A, X,\Omega, p}\right) \) lies at most \( n \) -sheeted over \( E \) if \( \# {p}^{-1}\left( \lambda \right) \leq n \) for each ... |
111_111_Three Dimensional Navier-Stokes Equations-James_C._Robinson,_Jos_L._Rodrigo,_Witold_Sadows(z-lib.org | Definition 14.4 |
Definition 14.4 We shall call the map \( \rho \left( {T}_{0}\right) \ni z \rightarrow \gamma \left( z\right) \in \mathbf{B}\left( {\mathcal{K},\mathcal{H}}\right) \) the gamma field and the map \( \rho \left( {T}_{0}\right) \ni z \rightarrow M\left( z\right) \in \mathbf{B}\left( \mathcal{K}\right) \) the Weyl functio... |
1063_(GTM222)Lie Groups, Lie Algebras, and Representations | Definition 4.20 |
Definition 4.20. Let \( G \) be a matrix Lie group and let \( {\Pi }_{1} \) and \( {\Pi }_{2} \) be representations of \( G \), acting on spaces \( {V}_{1} \) and \( {V}_{2} \) . Then the tensor product representation of \( G \) , acting on \( {V}_{1} \otimes {V}_{2} \), is defined by
\[
\left( {{\Pi }_{1} \otimes {... |
1068_(GTM227)Combinatorial Commutative Algebra | Definition 7.17 |
Definition 7.17 Let \( C \) be a rational pointed cone in \( {\mathbb{R}}^{d} \) . The pointed semigroup \( Q = C \cap {\mathbb{Z}}^{d} \) has a unique minimal generating set, called the Hilbert basis of the cone \( C \) and denoted by \( {\mathcal{H}}_{C} \) or \( {\mathcal{H}}_{Q} \), afforded by Theorem 7.16 and P... |
1079_(GTM237)An Introduction to Operators on the Hardy-Hilbert Space | Definition 2.2.9 |
Definition 2.2.9. A function \( \phi \in {\mathbf{H}}^{\infty } \) satisfying \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e. is an inner function.
Theorem 2.2.10. If \( \phi \) is a nonconstant inner function, then \( \left| {\phi \left( z\right) }\right| < 1 \) for all \( z \in \mathb... |
1057_(GTM217)Model Theory | Definition 3.1.1 |
Definition 3.1.1 We say that a theory \( T \) has quantifier elimination if for every formula \( \phi \) there is a quantifier-free formula \( \psi \) such that
\[
T \vDash \phi \leftrightarrow \psi
\]
We will start by showing that DLO, the theory of dense linear orders without endpoints, has quantifier elimination... |
1042_(GTM203)The Symmetric Group | Definition 3.8.4 |
Definition 3.8.4 A partial tableau \( P \) is miniature if \( P \) has exactly three elements. -
The miniature tableaux are used to model the dual Knuth relations of the first and second kinds.
Proposition 3.8.5 Let \( P \) and \( Q \) be distinct miniature tableaux of the same shape \( \lambda /\mu \) and content.... |
1234_[丁一文] Number Theory 1 | Definition 1.1.2 |
Definition 1.1.2. Let \( R \) be a commutative ring (with unit). We call \( R \) is euclidean if there exists \( f : R \rightarrow {\mathbb{Z}}_{ \geq 0} \) such that for all \( \alpha ,\beta \in R \) ,
1. \( f\left( \alpha \right) = 0 \Leftrightarrow \alpha = 0 \) ,
2. \( f\left( {\alpha \beta }\right) = f\left( \... |
18_Algebra Chapter 0 | Definition 4.4 |
Definition 4.4. An \( R \) -module \( M \) is cyclic if it is generated by a singleton, that is, if \( M \cong R/I \) for some ideal \( I \) of \( R \) .
The equivalence in the definition is hopefully clear to our reader, as an immediate consequence of the first isomorphism theorem for modules (Corollary 11115.16). ... |
1234_[丁一文] Number Theory 1 | Definition 3.1.3 |
Definition 3.1.3. Put \( \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} = \left\{ {\left( {x}_{j}\right) \mid {x}_{j} \in {G}_{j}}\right. \) and \( \left. {{x}_{j} \in {H}_{j}\text{for all but finitely many}j}\right\} \) , called the restricted direct product of \( {\left\{ {G}_{j}\right\} }_{j \in J} \) with re... |
1065_(GTM224)Metric Structures in Differential Geometry | Definition 1.1 |
Definition 1.1. Let \( \left( {{\xi }_{i},\langle ,{\rangle }_{i}}\right), i = 1,2 \), be Euclidean bundles over \( {M}_{i} \) . A map \( h : E\left( {\xi }_{1}\right) \rightarrow E\left( {\xi }_{2}\right) \) is said to be isometric if
(1) \( h \) maps each fiber \( {\pi }_{1}^{-1}\left( {p}_{1}\right) \) linearly i... |
117_《微积分笔记》最终版_by零蛋大 | Definition 4.13 |
Definition 4.13. An invertible measure-preserving transformation \( T \) of a probability space \( \left( {X,\mathcal{B}, m}\right) \) is a Kolmogorov automorphism \( \left( {K\text{-automorphism}}\right) \) if there exists a sub- \( \sigma \) -algebra \( \mathcal{K} \) of \( \mathcal{B} \) such that:
(i) \( \mathca... |
1077_(GTM235)Compact Lie Groups | Definition 2.30 |
Definition 2.30. If \( H \) is a Lie subgroup of a Lie group \( G \) and \( V \) is a representation of \( G \), write \( {\left. V\right| }_{H} \) for the representation of \( H \) on \( V \) given by restricting the action of \( G \) to \( H \) .
For the remainder of this section, view \( O\left( {n - 1}\right) \)... |
1068_(GTM227)Combinatorial Commutative Algebra | Definition 7.8 |
Definition 7.8 A subset \( T \subseteq Q \) is called an ideal of \( Q \) if \( Q + T \subseteq T \) . A subsemigroup \( F \) of \( Q \) is called a face if the complement \( Q \smallsetminus F \) is an ideal of \( Q \) . The affine semigroup \( Q \) is pointed if its only unit is \( \mathbf{0} \), where a unit is an... |
1059_(GTM219)The Arithmetic of Hyperbolic 3-Manifolds | Definition 0.7.1 |
Definition 0.7.1 The field \( K \) is said to be complete at \( v \) if every Cauchy sequence in \( K \) converges to an element of \( K \) .
For a number field \( k \), we have indicated how to obtain all valuations. The field \( k \) is not complete with respect to any of these valuations, but for each valuation \... |
1063_(GTM222)Lie Groups, Lie Algebras, and Representations | Definition 1.1 |
Definition 1.1. The general linear group over the real numbers, denoted \( \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \), is the group of all \( n \times n \) invertible matrices with real entries. The general linear group over the complex numbers, denoted \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \), is the group ... |
1112_(GTM267)Quantum Theory for Mathematicians | Definition 16.2 |
Definition 16.2 If \( {G}_{1} \) and \( {G}_{2} \) are matrix Lie groups, then a Lie group homomorphism of \( {G}_{1} \) to \( {G}_{2} \) is a continuous group homomorphism of \( {G}_{1} \) into \( {G}_{2} \) . A Lie group homomorphism is called a Lie group isomorphism if it is one-to-one and onto with continuous inv... |
1094_(GTM250)Modern Fourier Analysis | Definition 7.5.4 |
Definition 7.5.4. We denote by \( {\mathcal{S}}_{ * }\left( {\mathbf{R}}^{d}\right) \) the space of all Schwartz functions \( \Psi \) on \( {\mathbf{R}}^{d} \) whose Fourier transforms are supported in an annulus of the form \( {c}_{1} < \left| \xi \right| < {c}_{2} \) , are nonvanishing in a smaller annulus \( {c}_{... |
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