Misplaced Pages

#197802

31-462: The operation x ↦ ‖ x ‖ := d ( 0 , x ) {\displaystyle x\mapsto \|x\|:=d(0,x)} is called an F-norm , although in general an F-norm is not required to be homogeneous. By translation-invariance , the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with

62-407: A + q b and r a + s b for integers p , q , r , and s such that ps − qr is 1 or −1. This ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair. Each pair a , b defines a parallelogram, all with the same area, the magnitude of the cross product . One parallelogram fully defines

93-446: A complete F-norm. Some authors use the term Fréchet space rather than F-space , but usually the term "Fréchet space" is reserved for locally convex F-spaces. Some other authors use the term "F-space" as a synonym of "Fréchet space", by which they mean a locally convex complete metrizable topological vector space . The metric may or may not necessarily be part of the structure on an F-space; many authors only require that such

124-427: A physical system is equivalent to the momentum conservation law . Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group. Translational invariance implies that, at least in one direction,

155-428: A set of translation vectors is the hypervolume of the n -dimensional parallelepiped the set subtends (also called the covolume of the lattice). This parallelepiped is a fundamental region of the symmetry: any pattern on or in it is possible, and this defines the whole object. See also lattice (group) . E.g. in 2D, instead of a and b we can also take a and a − b , etc. In general in 2D, we can take p

186-760: A space be metrizable in a manner that satisfies the above properties. All Banach spaces and Fréchet spaces are F-spaces. In particular, a Banach space is an F-space with an additional requirement that d ( a x , 0 ) = | a | d ( x , 0 ) . {\displaystyle d(ax,0)=|a|d(x,0).} The L spaces can be made into F-spaces for all p ≥ 0 {\displaystyle p\geq 0} and for p ≥ 1 {\displaystyle p\geq 1} they can be made into locally convex and thus Fréchet spaces and even Banach spaces. L 1 2 [ 0 , 1 ] {\displaystyle L^{\frac {1}{2}}[0,\,1]}

217-502: Is finer or coarser than the other then they must be equal (that is, if τ ⊆ τ 2  or  τ 2 ⊆ τ  then  τ = τ 2 {\displaystyle \tau \subseteq \tau _{2}{\text{ or }}\tau _{2}\subseteq \tau {\text{ then }}\tau =\tau _{2}} ). Translation invariance In physics and mathematics , continuous translational symmetry

248-616: Is a complete topological vector space . The open mapping theorem implies that if τ  and  τ 2 {\displaystyle \tau {\text{ and }}\tau _{2}} are topologies on X {\displaystyle X} that make both ( X , τ ) {\displaystyle (X,\tau )} and ( X , τ 2 ) {\displaystyle \left(X,\tau _{2}\right)} into complete metrizable topological vector spaces (for example, Banach or Fréchet spaces ) and if one topology

279-637: Is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm). If p {\displaystyle p} is a quasinorm on X {\displaystyle X} then p {\displaystyle p} induces a vector topology on X {\displaystyle X} whose neighborhood basis at the origin is given by the sets: { x ∈ X : p ( x ) < 1 / n } {\displaystyle \{x\in X:p(x)<;1/n\}} as n {\displaystyle n} ranges over

310-447: Is a real-valued map p {\displaystyle p} on X {\displaystyle X} that satisfies the following conditions: A quasinorm is a quasi-seminorm that also satisfies: A pair ( X , p ) {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} and an associated quasi-seminorm p {\displaystyle p}

341-446: Is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space . Let W p ( D ) {\displaystyle W_{p}(\mathbb {D} )} be the space of all complex valued Taylor series f ( z ) = ∑ n ≥ 0 a n z n {\displaystyle f(z)=\sum _{n\geq 0}a_{n}z^{n}} on

SECTION 10

#1732884827198

372-832: Is called a quasi-Banach algebra . A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin. Since every norm is a quasinorm, every normed space is also a quasinormed space. L p {\displaystyle L^{p}} spaces with 0 < p < 1 {\displaystyle 0<p<1} The L p {\displaystyle L^{p}} spaces for 0 < p < 1 {\displaystyle 0<p<1} are quasinormed spaces (indeed, they are even F-spaces ) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology). For 0 < p < 1 , {\displaystyle 0<p<1,}

403-422: Is called a quasi-seminormed vector space . If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space . Multiplier The infimum of all values of k {\displaystyle k} that satisfy condition (3) is called the multiplier of p . {\displaystyle p.} The multiplier itself will also satisfy condition (3) and so it

434-546: Is called a quasinormed algebra if the vector space A {\displaystyle A} is an algebra and there is a constant K > 0 {\displaystyle K>0} such that ‖ x y ‖ ≤ K ‖ x ‖ ⋅ ‖ y ‖ {\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|} for all x , y ∈ A . {\displaystyle x,y\in A.} A complete quasinormed algebra

465-426: Is the invariance of a system of equations under any translation (without rotation ). Discrete translational symmetry is invariant under discrete translation. Analogously, an operator A on functions is said to be translationally invariant with respect to a translation operator T δ {\displaystyle T_{\delta }} if the result after applying A doesn't change if

496-459: Is the unique smallest real number that satisfies this condition. The term k {\displaystyle k} -quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to k . {\displaystyle k.} A norm (respectively, a seminorm ) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is 1. {\displaystyle 1.} Thus every seminorm

527-580: The Lebesgue space L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} is a complete metrizable TVS (an F-space ) that is not locally convex (in fact, its only convex open subsets are itself L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} and the empty set) and the only continuous linear functional on L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])}

558-562: The p-norm : ‖ f ‖ p = ∑ n | a n | p ( 0 < p < 1 ) . {\displaystyle \|f\|_{p}=\sum _{n}\left|a_{n}\right|^{p}\qquad (0<p<1).} In fact, W p {\displaystyle W_{p}} is a quasi-Banach algebra . Moreover, for any ζ {\displaystyle \zeta } with | ζ | ≤ 1 {\displaystyle |\zeta |\leq 1}

589-444: The argument function is translated. More precisely it must hold that ∀ δ   A f = A ( T δ f ) . {\displaystyle \forall \delta \ Af=A(T_{\delta }f).} Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem , space translational symmetry of

620-410: The dimension. This implies that the object is infinite in all directions. In this case, the set of all translations forms a lattice . Different bases of translation vectors generate the same lattice if and only if one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1. The absolute value of the determinant of the matrix formed by

651-461: The map f ↦ f ( ζ ) {\displaystyle f\mapsto f(\zeta )} is a bounded linear (multiplicative functional) on W p ( D ) . {\displaystyle W_{p}(\mathbb {D} ).} Theorem   (Klee (1952))  —  Let d {\displaystyle d} be any metric on a vector space X {\displaystyle X} such that

SECTION 20

#1732884827198

682-427: The norm axioms, except that the triangle inequality is replaced by ‖ x + y ‖ ≤ K ( ‖ x ‖ + ‖ y ‖ ) {\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)} for some K > 1. {\displaystyle K>1.} A quasi-seminorm on a vector space X {\displaystyle X}

713-404: The object is infinite: for any given point p , the set of points with the same properties due to the translational symmetry form the infinite discrete set { p + n a | n ∈ Z } = p + Z a . Fundamental domains are e.g. H + [0, 1] a for any hyperplane H for which a has an independent direction. This is in 1D a line segment , in 2D an infinite strip, and in 3D a slab, such that

744-447: The other translation vector starting at one side of the rectangle ends at the opposite side. For example, consider a tiling with equal rectangular tiles with an asymmetric pattern on them, all oriented the same, in rows, with for each row a shift of a fraction, not one half, of a tile, always the same, then we have only translational symmetry, wallpaper group p 1 (the same applies without shift). With rotational symmetry of order two of

775-420: The pattern on the tile we have p 2 (more symmetry of the pattern on the tile does not change that, because of the arrangement of the tiles). The rectangle is a more convenient unit to consider as fundamental domain (or set of two of them) than a parallelogram consisting of part of a tile and part of another one. In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in

806-516: The positive integers. A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space . Every quasinormed topological vector space is pseudometrizable . A complete quasinormed space is called a quasi-Banach space . Every Banach space is a quasi-Banach space, although not conversely. A quasinormed space ( A , ‖ ⋅ ‖ ) {\displaystyle (A,\|\,\cdot \,\|)}

837-408: The same direction, fully defines the whole object. Similarly, in 3D there may be translational symmetry in one or two directions for vectors of any length. One plane ( cross-section ) or line, respectively, fully defines the whole object. Quasi-Banach algebra In linear algebra , functional analysis and related areas of mathematics , a quasinorm is similar to a norm in that it satisfies

868-476: The topology τ {\displaystyle \tau } induced by d {\displaystyle d} on X {\displaystyle X} makes ( X , τ ) {\displaystyle (X,\tau )} into a topological vector space. If ( X , d ) {\displaystyle (X,d)} is a complete metric space then ( X , τ ) {\displaystyle (X,\tau )}

899-469: The unit disc D {\displaystyle \mathbb {D} } such that ∑ n | a n | p < ∞ {\displaystyle \sum _{n}\left|a_{n}\right|^{p}<\infty } then for 0 < p < 1 , {\displaystyle 0<p<1,} W p ( D ) {\displaystyle W_{p}(\mathbb {D} )} are F-spaces under

930-425: The vector starting at one side ends at the other side. Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector. In spaces with dimension higher than 1, there may be multiple translational symmetries. For each set of k independent translation vectors, the symmetry group is isomorphic with Z . In particular, the multiplicity may be equal to

961-491: The whole object. Without further symmetry, this parallelogram is a fundamental domain. The vectors a and b can be represented by complex numbers. For two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the modular group , see lattice (group) . Alternatively, e.g. a rectangle may define the whole object, even if the translation vectors are not perpendicular, if it has two sides parallel to one translation vector, while

F-space - Misplaced Pages Continue

#197802