Schaefer, H. H., 1999, Topological Vector Spaces, 2nd ed. Springer, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... #, defined by ## : # [0,#) can be extended to distributions, by restricting their actions to those C # functions whose supports are contained in [0, #) That is, ###,## ##, ##, supp # #) 1) The space # : # n 0 L [ n, 0] with the inductive limit topology (see Schaefer [1971], Treves [1967] is called the space of inputs, and # : L , i.e. the p product of the space of locally Lebesgue square integrable functions on [0, #) is called the space of outputs. For a locally square integrable function #, ### [a,b] denotes its L norm on [a, b] With ....

.... is enough to show that M N #) Note that the closure here must be taken in the space L (see (7) so that M N is not obvious even though M and N are contained in L #) Recall that for a subspace L in a Banach space X, its polar L # is the subspace of X # defined by (Schaefer [1971]) L # : x X # ; # ,x# = 0 for all x L . This is an analog of the notion of the orthogonal complements in Hilbert spaces. The di#erence is that in discussing polars we fix the duality #X, X # #, so when we take the polar of a subset in X # , we consider it in X rather than in X ## . ....

H. H. Schaefer, Topological Vector Spaces, Springer, 1971.

....graph, and hence it is continuous. We now note that (X # (R ) This is because C # [0, # (R ) with respect to the duality ##,# #,##E # (R ) # where # (t) # ( t) Then the polar of the (X is easily seen to be equal to q # (R ) and hence (X # (R ) by [3]. This yields the continuity of # (R ) p# # E # (R ) and also its invertibility. Then it follows that [p#b] #]inE # (R ) q # (R ) i.e. p#b mod q, i.e. p b # = q a for some a # (R ) This implies q ( a) p b = #. That is, p, q) is exactly coprime. # Remark ....

H. H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York, 1971.

.... defined by ## : # [0,#) can be extended to distributions, by restricting their actions to those C # functions whose supports are contained in [0, #) That is, ###,## ##, ##, supp # #) 2) The space # : # n 0 L [ n, 0] with the inductive limit topology (see Schaefer [S1], Treves [T] is called the space of inputs, and # : L loc [0, #) i.e. the p product of the space of locally Lebesgue square integrable functions on [0, #) is called the space of outputs. With respect to the family of L norms on all bounded 4 intervals, # is a Frechet ....

H. H. Schaefer, Topological Vector Spaces, Springer, 1971.

....chapter II, section 2) Recall the wellknown fact (see, e.g. KPR 84] theorem 2. 2) that, for a diffuse measure P, the topological dual of L ; F ; P) is reduced to f0g so that there is no counterpart to the duality theory, which works so nicely in the context of locally convex spaces (compare [Sch 67] chapter IV) By L ; F ; P) we denote the positive orthant of L ; F ; P) i.e. F ; P) ff 2 L ; F ; P) f 0g: The research of this paper was financially supported by the Austrian Science Foundation (FWF) under grant SFB#10 ( Adaptive Information Systems and Modelling in ....

....in the next section. The proof of theorem 1.3 will be given in section 3. We finish this introductory section by giving an easy extension of the bipolar theorem 1. 3 to subsets of L (as opposed to subsets of L ) Recall that, with the usual definition of solid sets in vector lattices (see [Sch 67] chapter V, section 1) a set D ae L is defined to be solid in the following way. 1.4 Definition. A set D ae L is solid, if f 2 D and h 2 L with jhj jf j implies h 2 D. Note that a set D ae L is solid if and only if the set of its absolut values jDj = fjhj : h 2 Dg ae L form a ....

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. Schaefer, H.H. (1966), Topological Vector Spaces, Springer Graduate Texts in Mathematics.

....L 1 (Q ) closure of the vector space generated by G 0 is contained in f Delta S T : 2 L 1 loc (S; Q ) such that Delta S is a Q martingaleg: According to Proposition 1.1 in [13] this result is valid without the assumption of a complete filtration. Extending Proposition I.3. 3 in [25] from vector spaces to the class of closed convex cones one gets f 0 i dQ dP j = d Delta ST n X i=1 i H i Q a.s. where i 0 for 1 i r. This implies by the definition of d that i (EQ H i ) 0 for 1 i r. The following theorem is a variant of Theorem 3.1. It shows that ....

....in variation. As H 1 ; Hn are bounded this is also true for M 0 : fQ 2 M : EQH i = 0 for r 1 i ng. Since M, M 0 are convex, they are also closed in oe(L 1 ; L 1 ) if one identifies Q 2 M with its Radon Nikodym density with respect to P (see for example Proposition IV.3. 1 in [25]) We define a function B : M 0 R r by B(Q) GammaE QH 1 ; GammaE QH r ) Obviously the component mappings of B are convex and continuous with respect to oe(L 1 ; L 1 ) The optimization problem is to minimize the lower semicontinuous functional f( DeltajjP ) see Theorem ....

H.H. Schaefer. Topological Vector Spaces. Springer, Berlin, 1971.

....theory fornonnegativ e matrices (see [6, 9, 15, 21] for a completesurv ey) to a larger class of linear transformations. As a result, anextensiv e literature on the subject is now av ailable. In fact, the infinite dimensional case, firstdev eloped by Krein and Rutman in [14] is fully discussed in [17, 18], while the finite dimensional aspects of this theory can be found in [6] In the finite dimensional context, research e#orts led to the introduction of the notion of a matrix that leav es a proper cone inv ariant [2, 3, 7, 24] and to the determination of necessary and su#cient conditions for a ....

H.H. Schaefer, Topological Vector Spaces, 4th ed., Springer, NewYork, 1980.

....convex spaces. In particular, D r (M; X) is an LF space provided every C r K (M; X) is a Fr echet space, and an LB space provided every C r K (M; X) is a Banach space. Furthermore, we deduce that the inclusion maps C r K (M; X) D r (M; X) are topological embeddings for all K 2 K(M) [34], Chapter II, x6, Assertion 6.3) To obtain a more explicit description of the topology on D r (M; X) we adapt ideas from ( 35] Chapter III, x1) and [1] De nition 4.7 Given any sequences q = q n ) n2N , k = k n ) n2N , and e = n ) n2N of seminorms q n 2 , natural numbers k n 2 jr] ....

Schaefer, H. H., \Topological Vector Spaces," Springer-Verlag, 1971.

....continuous on each S 2 S, is continuous on (E; f) GammaE; S ) is BB reflexive, i.e. it is bicontinuously isomorphic to Gamma c Gamma c ( GammaE; S ) Proof. The equivalence between a) and b) is properly Grothendieck Theorem. The proof can be seen in any classical treatise, for example [21]. In [8] it is proved that a locally convex vector space is complete if and only if it is BB reflexive as a vector space, thus a) c) 9 b) e) is precisely (ii) of lemma 3.1 c) f) and a) d) are obtained through the topological isomorfism ae : LSE Gamma SE (Lemma 3.2 ) Remark For any ....

.... GammaE which is ( GammaE; E) continuous on every equicontinuous subset of GammaE , is ( GammaE; E) continuous on all GammaE . d) E is BB reflexive as a topological vector space. e) E is BB reflexive as a topological group. Proof. a) b) is a standard corollary of GT, see for example [21]. a) d) and (d) e) are proved in [8] and [9] respectively. c) e) In order to see that E is a topological isomorphism, only surjectivity is to be seen, since E is already an embedding ( lemma 3.6) Let 2 Gamma Gamma c E. If H ae GammaE is equicontinuous, jH is ( GammaE; ....

Schaefer,H.H. Topological Vector Spaces. Graduate Texts in Mathematics 3. SpringerVerlag 1970.

....generally, random elements with values in the topological dual to a Frech et nuclear space (or to the strict inductive limit of a sequence of Frech et nuclear spaces) For the sake of brevity we will formulate here results for the simpler case only. Let Phi be a Frech et nuclear space (see e.g. [10]) Let k Deltak 1 k Deltak 2 : be an increasing sequence of Hilbertian seminorms defining the topology on Phi. Denote by ( Phi p ; k Delta k p ) the Hilbert space arising by completion of the quotient space Phi=k Delta k p and by ( Phi 0 Gammap ; k Delta k Gammap ) the topological ....

Schaefer, H.H., Topological Vector Spaces, Springer, Berlin 1970.

....or, more generally, random elements with values in the topological dual to a Frech et nuclear space (or to the strict inductive limit of a Frech et nuclear spaces) For the sake of brevity we will formulate here results for the simpler case only. Let Phi be a Frech et nuclear space (see e.g. [28]) Let k Delta k 1 k Delta k 2 : be an increasing sequence of Hilbertian seminorms defining the topology on Phi. Denote by ( Phi p ; k Delta k p ) the Hilbert space arising by completion of the quotient space Phi=k Delta k p and by ( Phi 0 Gammap ; k Delta k Gammap ) the topological ....

Schaefer, H.H., Topological Vector Spaces, Springer, Berlin 1970.

....any 2 ] The operator L jqj 2 will be used to estimate the regularity of . In our proofs we shall use that L jqj 2 is a positive operator, in the sense that (L jqj 2f) 0 for all 2 [ if f( 0 for all 2 [ Such operators have special spectral properties; see e.g. [31] or [32] To see how the general theorems on positive operators apply here, we rst need to establish some facts about E . De ne E = ff 2 E ; f( 0 for 2 [ g. This is a cone in E , which contains in particular all the positive trigonometric polynomials. It follows that the ....

....here, we rst need to establish some facts about E . De ne E = ff 2 E ; f( 0 for 2 [ g. This is a cone in E , which contains in particular all the positive trigonometric polynomials. It follows that the closed linear span of E equals E , or in the terminology of Schae er [31], E is an ordered Banach space with total positive cone. It then already follows from the KreinRutman theorem (see e.g. 31] p. 265) that 12 I. Daubechies Lemma 4.2 The spectral radius r of L jqj 2 in E is an eigenvalue for L jqj 2 and there exists a positive eigenfunction for this ....

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Schaefer, H., Topological vector spaces, MacMillan, New York, 1966.

....of this result without the assumption of a complete filtration see Jacod (1979, Proposition 1.1) on the closedness of stochastic integrals L 1 (G; Q ) is contained in f Delta S T : 2 L 1 loc (S) such that Delta S is a martingaleg. By a simple closedness argument (see for example Schaefer (1971), pg. 22) we are done. The following Theorem is a variant of Theorem 3.1. It shows that the necessary condition in Theorem 3.1 is also valid for the set of local martingale measures under the additional assumption that the price process is locally bounded. For the case of the relative entropy it ....

Schaefer, H. (1971). Topological Vector Spaces. Berlin: Springer.

....[GHOR00] Let N be a complex Fr echet nuclear space with topology given by an increasing family of Hilbertian norms fj Deltaj n ; n 2 Ng. It is well known that N may be represented as N = n2N N n , where the Hilbert space N n is the completion of N with respect to j Delta j n , see e.g. Sch71] GV68] By the general duality theory N 0 is given by N 0 = n2N N Gamman , where N Gamman = N 0 n is the topological dual of N n . Let : R R be a continuous 1 convex strictly increasing function such that lim x 1 (x) x = 1; 0) 0: Such functions are called Young ....

H. H. Schaefer. Topological Vector Spaces. Springer-Verlag, Berlin, Heidelberg and New York, 1971.

....methods are commonly used to obtain a problem with better analytical properties in the areas of Calculus of variations, control theory and differential games. For the minimax problem, we have adapted some results contained in [11, 12, 14, 15] and used techniques and concepts which can be seen in [6, 10, 13]. We want to remark that a similar procedure has been employed by Barron and Jensen in [3] for the finite horizon case. Acknowledgements The authors would like to thank: ffl Laura S. Aragone for their careful revision of the manuscript. ffl CONICET for support given through the grant PID ....

Schaefer H.H., Topological vector spaces, Springer--Verlag, New York, 1980. RR n2945 24 Silvia C. Di Marco and Roberto L.V. Gonz'alez

.... Gamma p;m (n) and that its image splits. This last condition arises because we are embedding Sigma Gamma p;m (n) in an infinite dimensional 11 space [44] but is automatically satisfied here since ImDj q (W ) is finite dimensional hence splits in the Banach space H p Thetam 1 (see e.g. [48], 51] Letting (A; B; C; D) be a minimal realization of W , the image of Dj q (W ) obviously contains the image of the derivative of j q ffi Pi at (A; B; C; D) If we differentiate this map, which is formally given by (2.1) with respect to the arguments, we find that D 4 (j q ffi Pi) C ....

H.H. Schaefer. Topological vector spaces. Springer, New--York, 1986.

....If f 2 C 1( Omega Gamma then Df denotes its (total) derivative. Also, we set f (x) f( Gammax) On any cartesian product, pr i denotes the projection onto the i th factor. For r 2 R, r] is the largest integer r. We set I = 0; 1] Concerning locally convex spaces our basic reference is [39]. In particular, by a locally convex space we mean a vector space endowed with a locally convex Hausdorff topology. The space of test functions (i.e. compactly supported smooth functions) on Omega is denoted by D( Omega Gamma and is equipped with its natural (LF) topology; its dual, the space ....

H. H. Schaefer, Topological Vector Spaces (5th ed.), Grad. Texts in Math., Springer, 1986.

....of Theorem 8 be satisfied, let B be dense in C, and let d 2 int(A[C] Then B A Gamma1 (d) is dense in C A Gamma1 (d) To apply this result in a concrete situation, the openness of the map Aj S with respect to has to be ensured, i.e. an open mapping theorem is needed. We refer to [15] for a collection of such results. It is known that the open mapping theorem holds true for F spaces, that is, for complete, metrizable topological vector spaces [15, p. 77] The resulting statement, as used in our examples, is formulated in the following theorem. For ease of reference, we ....

H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.

.... S and T on a metric linear space X and show that if the sequence of Ishikawa iterates associated with S and T converges, it coverges to a common fixed point of S and T: In the sequel we assume that the topology on X is generated by an F norm q which has the following properties (see [5], pp.28 29) a) q(x) 0 and q(x) 0 iff x = 0 (b) q(x y) q(x) q(y) c) q(ax) q(x) for all (real or complex) scalars a with jaj 1 (d)If an a and xn x, then q(a n xn Gamma ax) 0. Note that q is continuous on X; the equation d(x; y) q(x Gamma y) defines a translation invariant ....

H.H.Schaefer, Topological vector spaces, Spriner-Verlag,Berlin 1971.

....oe(F; G) The space (F; oe(F; G) is locally convex and is Hausdorff if and only if the duality is separated in F (condition (S 1 ) Any topology consistent with the duality is stronger than the weak topology and weaker than the Mackey topology. More details and all the results can be found in [12]. They motivate why the notion of separated and or extensional Chu spaces on Ban 1 is interesting. 3. A group algebra Now we have all the ingredients to define an algebra in chu associated to a Hausdorff group, such that there exists a bijection between modules over this algebra and isometric ....

H. H. Schaefer, Topological vector spaces, Macmillan, New York, 1966

....generated by Y 0 ; y 1 and y 2 . We shall consider the dual pair hX; Y i equipped with its natural scalar product h Delta; Deltai and we define on X the weak topology = oe(X; Y ) generated by the space Y (for a definition of the weak topology defined for a dual pair of vector spaces see, e.g. Sch 66] Observe that this topology is just slightly finer than the product topology on l 1 which was used by D. Kreps in ( K 81] ex. 4.3) indeed, the product topology on l 1 equals the weak topology oe(X; Y 0 ) also note that Y 0 is a subspace of codimension 2 of Y . The cone K will be ....

. Schaefer, H.H. (1966), Topological Vector Spaces, Springer Graduate Texts in Mathematics.

....; 2.1) where H is a Hilbert space, Phi a dense subspace of H, endowed with a locally convex topology Phi , finer than the norm topology inherited from H (i.e. a stronger notion of convergence) and Phi Theta is the space of Phi continuous antilinear functionals on Phi. By duality [49, 57], each space in (2.1) is dense in the next one and all embeddings are linear and continuous. Notice that Phi Theta is determined uniquely by Phi, but that there are two ways of obtaining the couple Phi; H, namely: ffl Either one starts from H and builds Phi, for instance as the largest ....

....requires that Phi satisfy a number of properties: ffl Phi is complete with respect to Phi , that is, every Cauchy (generalized) sequence converges to an element of Phi. ffl Phi is reflexive, that is, Phi Theta ) Theta Phi, if Phi Theta is given its strong dual topology [49, 57]. ffl In most cases, Phi is obtained as the intersection of a countable family of Hilbert spaces, Phi = n2N H n . It is then a Fr echet space. ffl Phi is nuclear; in the case where Phi = n2N H n , this means that, for each n, there is m n such that the embedding Hm H n is a ....

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H.H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971

.... in H we can construct the nuclear triple N ae H ae N 0 : The dual pairing h Delta; Deltai of N and N 0 is then realized as an extension of the inner product in H: hf; i = f; f 2 H; 2 N : Instead of reproducing the abstract definition of nuclear spaces, see e.g. Pie69] and [Sch71] we give a complete and convenient characterization in terms of projective limits of Hilbert spaces which has also been proved in [Pie69] and in [Sch71] Theorem 2.1 The nuclear Fr echet space can be represented as N = p2N H p ; where f(H p ; Delta; Delta) p ) p 2 Ng is a family of ....

.... inner product in H: hf; i = f; f 2 H; 2 N : Instead of reproducing the abstract definition of nuclear spaces, see e.g. Pie69] and [Sch71] we give a complete and convenient characterization in terms of projective limits of Hilbert spaces which has also been proved in [Pie69] and in [Sch71] Theorem 2.1 The nuclear Fr echet space can be represented as N = p2N H p ; where f(H p ; Delta; Delta) p ) p 2 Ng is a family of Hilbert spaces such that for all p 1 ; p 2 2 N there exists p 2 N such that the embeddings H p , H p 1 and H p , H p 2 are Hilbert Schmidt. The ....

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H.H. Schaefer. Topological Vector Spaces. Springer Verlag, Berlin, Heidelberg, New York, 1971.

....II, section 2) Recall the wellknown fact (see, e.g. KPR 84] theorem 2. 2) that, for a diffuse measure P, the topological dual of L 0( Omega ; F ; P) is reduced to f0g so that there is no counterpart to the duality theory, which works so nicely in the context of locally convex spaces (compare [Sch 67] chapter IV) By L 0 ( Omega ; F ; P) we denote the positive orthant of L 0( Omega ; F ; P) i.e. L 0 ( Omega ; F ; P) ff 2 L 0( Omega ; F ; P) f 0g: The research of this paper was financially supported by the Austrian Science Foundation (FWF) under grant SFB#10 ( Adaptive ....

....in the next section. The proof of theorem 1.3 will be given in section 3. We finish this introductory section by giving an easy extension of the bipolar theorem 1. 3 to subsets of L 0 (as opposed to subsets of L 0 ) Recall that, with the usual definition of solid sets in vector lattices (see [Sch 67] chapter V, section 1) a set D ae L 0 is defined to be solid in the following way. 1.4 Definition. A set D ae L 0 is solid, if f 2 D and h 2 L 0 with jhj jf j implies h 2 D. Note that a set D ae L 0 is solid if and only if the set of its absolut values jDj = fjhj : h 2 Dg ae L 0 ....

[Article contains additional citation context not shown here]

. Schaefer, H.H. (1966), Topological Vector Spaces, Springer Graduate Texts in Mathematics.

....H is a real separable Hilbert space containing S(R d ) as a dense and topological subspace. For instance, H can be chosen as the space of real valued square integrable functions w.r.t. the Lebesgue measure on R d or as a Sobolev space on R d . As is well known, see e.g. Pie72] and [Sch71] S = S(R d ) is the projective limit of a family of Hilbert spaces (H p ) p2N0 ; H 0 = H, such that for all p 1 ; p 2 2 N there exists p 2 N such that H p ae H p1 and H p ae H p2 and the embeddings are of Hilbert Schmidt class. i.e. S is a countably Hilbert space in the sense of [GV68] ....

....we introduce the notion of symmetric tensor power of the nuclear space S. The simplest way to do this is to start from usual symmetric tensor powers H Omega n p ; n 2 N , of Hilbert spaces. Using the definition S Omega n : prlim p2N H Omega n p one can prove, see e. g [Pie72] and [Sch71] that S Omega n is a nuclear space which is called the n th symmetric tensor power of S. The dual space S 0 Omega n can be written as S 0 Omega n = indlim p2N H Omega n Gammap : The space S 0 (R d ) Omega n is canonically isomorphic to S 0 (R nd ) the space ....

H.H. Schaefer. Topological Vector Spaces. Springer Verlag, Berlin, Heidelberg, New York, 1971.

.... topologically embedded in H we can construct the nuclear triple N ae H ae N 0 : The dual pairing h Delta; Deltai of N 0 and N then is realized as an extension of the inner product in H hf; i = f; f 2 H; 2 N Instead of reproducing the abstract definition of nuclear spaces (see e.g. [Sch71]) we give a complete (and convenient) characterization in terms of projective limits of Hilbert spaces. 2 PRELIMINARIES 3 Theorem 1 The nuclear Fr echet space N can be represented as N = p2IN H p , where fH p , p 2 INg is a family of Hilbert spaces such that for all p 1 ; p 2 2 IN there ....

....powers H Omega n p ; n 2 IN of Hilbert spaces. Since there is no danger of confusion we will preserve the notation j Deltaj p and j Deltaj Gammap for the norms on H Omega n p and H Omega n Gammap respectively. Using the definition N Omega n : pr lim p2IN H Omega n p one can prove [Sch71] that N Omega n is a nuclear space which is called the n th tensor power of N : The dual space of N Omega n can be written Gamma N Omega n Delta 0 = ind lim p2IN H Omega n Gammap Most important for the applications we have in mind is the following kernel theorem , see e.g. ....

Schaefer, H.H. (1971), Topological Vector Spaces, Springer, New York.

....that there exists a positive number d 0 which satisfies the inequality ( y y y y = 0 0 d a contradiction. q.e.d. Lemma 1 guarantees that the Banach space Y is a partially ordered space by means of the cone K. Note that, taking into account Krein s Theorem (see, for example, [17, 18] or [21] we can make the following observation. Conclusion. Suppose that the Banach space Y is partially ordered by means of the normal cone K. Then the equality Y =K K , 1) holds, where Y is a conjugate space for the space Y, and K is a conjugate (dual) cone of the cone ....

Schaefer, H. H., Topological Vector Spaces, Springer, New York, 1971.

.... embedded in H we can construct the nuclear triple N ae H ae N 0 : The dual pairing h Delta; Deltai of N 0 and N then is realized as an extension of the inner product in H hf; i = f; f 2 H; 2 N : Instead of reproducing the abstract definition of nuclear spaces (see e.g. [Sch71]) we give a complete (and convenient) characterization in terms of projective limits of decreasing chains of Hilbert spaces H p ; p 2 IN. Theorem 1 The nuclear Fr echet space N can be represented as N = p2IN H p ; where fH p ; p 2 INg is a family of Hilbert spaces such that for all p 1 ; p ....

....powers H Omega n p ; n 2 IN of Hilbert spaces. Since there is no danger of confusion we will preserve the notation j Deltaj p and j Deltaj Gammap for the norms on H Omega n p and H Omega n Gammap respectively. Using the definition N Omega n : pr lim p2IN H Omega n p ; one can prove [Sch71] that N Omega n is a nuclear space which is called the n th tensor power of N . The dual space of N Omega n can be written N0 Omega n = ind lim p2IN H Omega n Gammap : We also want to introduce the (Boson or symmetric) Fock space Gamma(H) of H by Gamma(H) 1 M n=0 H Omega ....

Schaefer, H.H. (1971), Topological Vector Spaces. Springer, New York.

....; 2.1) where H is a Hilbert space, Phi a dense subspace of H, endowed with a locally convex topology Phi , finer than the norm topology inherited from H (i.e. a stronger notion of convergence) and Phi Theta is the space of Phi continuous antilinear functionals on Phi. By duality [29, 33], each space in (2.1) is dense in the next one and all embeddings are linear and continuous. Notice that Phi Theta is determined uniquely by Phi, but that there are two ways of obtaining the couple Phi; H, namely: ffl Either one starts from H and builds Phi, for instance as the largest ....

....requires that Phi satisfy a number of properties: ffl Phi is complete with respect to Phi , that is, every Cauchy (generalized) sequence converges to an element of Phi. ffl Phi is reflexive, that is, Phi Theta ) Theta Phi, if Phi Theta is given its strong dual topology [29, 33]. ffl In most cases, Phi is obtained as the intersection of a countable family of Hilbert spaces, Phi = n2N H n . It is then a Fr echet space. ffl Phi is nuclear; in the case where Phi = n2N H n , this means that, for each n, there is m n such that the embedding Hm H n is a ....

[Article contains additional citation context not shown here]

H.H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971

....2.1 Gaussian spaces We start by considering a standard Gel fand triple N ae H ae N 0 : H is a real separable Hilbert space with inner product ( Delta; Delta) and norm j Delta j and N is a separable nuclear space densely topologically embedded in H. As is well known, see e.g. Pie72] and [Sch71] N is the projective limit of a family of Hilbert spaces (H p ) p2N , such that for all p 1 ; p 2 2 N there exists p 2 N such that H p ae H p1 and H p ae H p2 and the embeddings are of HilbertSchmidt class. i.e. N is a countably Hilbert space in the sense of [GV68] The dual space space N 0 ....

....we introduce the notion of symmetric tensor power of a nuclear space. The simplest way to do this is to start from usual symmetric tensor powers H Omega n p ; n 2 N , of Hilbert spaces. Using the definition N Omega n : prlim p2N H Omega n p one can prove, see e. g [Pie72] and [Sch71] that N Omega n is a nuclear space which is called the n th symmetric tensor power of N . The dual space N 0 Omega n can be written as N 0 Omega n = indlim p2N H Omega n Gammap : All the results quoted above also hold for complex spaces. In addition, we introduce the ....

H.H. Schaefer. Topological Vector Spaces. Springer Verlag, Berlin, Heidelberg, New York, 1971.

.... in H we can construct the nuclear triple N ae H ae N 0 : The dual pairing h Delta; Deltai of N and N 0 is realized then as an extension of the inner product in H: hf; i = f; f 2 H; 2 N : Instead of reproducing the abstract definition of nuclear spaces, see e.g. Pi69] and [Sc71], we give a complete and convenient characterization in terms of projective limits of Hilbert spaces which has also been proved in [Pi69] and in [Sc71] Theorem 2.1 The nuclear Fr echet space can be represented as N = p2N H p ; where f(H p ; Delta; Delta) p ) p 2 Ng is a family of ....

.... of the inner product in H: hf; i = f; f 2 H; 2 N : Instead of reproducing the abstract definition of nuclear spaces, see e.g. Pi69] and [Sc71] we give a complete and convenient characterization in terms of projective limits of Hilbert spaces which has also been proved in [Pi69] and in [Sc71]. Theorem 2.1 The nuclear Fr echet space can be represented as N = p2N H p ; where f(H p ; Delta; Delta) p ) p 2 Ng is a family of Hilbert spaces such that for all p 1 ; p 2 2 N there exists p 2 N such that the embeddings H p , H p 1 and H p , H p 2 are Hilbert Schmidt. The topology ....

[Article contains additional citation context not shown here]

Schaefer, H.H. (1971), Topological Vector Spaces, Springer, New York.

....exp(ngN )dP n lim sup n 1 1 n log i P n (B x;2 ) exp( Gamman N) j (28) maxf GammaI(B x;2 ) GammaN g Since (P n ) n is exponentially tight by Lemma 2.6 [17] we can choose a large compact set K. Then K : K [ fxg is compact, too. By the Stone Weierstrass theorem (see e.g. [20]) there is a finite collection fg i g 1in of functions in G such that sup y2K jg N (y) Gamma max i g i (y)j : Moreover, since g N (y) 0, passing to g i 0 if necessary, we assume g i 0 for all i and we can choose g 0 = 0. Without loss of generality let max i g i (y) 0 for all y 2 B x; ....

Schaefer, H.: Topological Vector Spaces, Springer--Verlag, New--York (1971)

....2.1 Preliminaries We start by considering a standard Gel fand triple N ae H ae N 0 : H is a real separable Hilbert space with inner product ( Delta; Delta) and norm j Delta j and N is a separable nuclear space densely topologically embedded in H. As is well known, see e.g. Pie69] and [Sch71] N is the projective limit of a family of Hilbert spaces (H p ) p2N , such that for all p 1 ; p 2 2 N there exists p 2 N such that H p ae H p 1 and H p ae H p 2 and the embeddings are of HilbertSchmidt class. i.e. N is a countably Hilbert space in the sense of [GV68] The dual space space N ....

....we introduce the notion of symmetric tensor power of a nuclear space. The simplest way to do this is to start from usual symmetric tensor powers H Omega n p ; n 2 N, of Hilbert spaces. Using the definition N Omega n : prlim p2N H Omega n p one can prove, see e. g [Pie69] and [Sch71] that N Omega n is a nuclear space which is called the n th symmetric tensor power of N . The dual space N 0 Omega n can be written as N 0 Omega n = indlim p2N H Omega n Gammap : All the results quoted above also hold for complex spaces. The symmetric (or Boson) Fock ....

H.H. Schaefer. Topological Vector Spaces. Springer Verlag, Berlin, Heidelberg, New York, 1971.

....2.1 Preliminaries We start by considering a standard Gel fand triple N ae H ae N 0 : H is a real separable Hilbert space with inner product ( Delta; Delta) and norm j Delta j and N is a separable nuclear space densely topologically embedded in H. As well known, see e.g. Pie69] and [Sch71] N is the projective limit of a family of Hilbert spaces (H p ) p2N , such that for all p 1 ; p 2 2 N there exists p 2 N such that H p ae H p 1 and H p ae H p2 and the embeddings are of Hilbert Schmidt class. i.e. N is a countably Hilbert space in the sense of [GV68] The dual space space N 0 ....

....we introduce the notion of symmetric tensor power of a nuclear space. The simplest way to do this is to start from usual symmetric tensor powers H Omega n p ; n 2 N , of Hilbert spaces. Using the definition N Omega n : prlim p2N H Omega n p one can prove, see e. g [Pie69] and [Sch71] that N Omega n is a nuclear space which is called the n th symmetric tensor power of N . The dual space N 0 Omega n can be written as N 0 Omega n = indlim p2N H Omega n Gammap : All the results quoted above also hold for complex spaces. The symmetric (or Boson) Fock space ....

H.H. Schaefer. Topological Vector Spaces. Springer Verlag, Berlin, Heidelberg, New York, 1971.

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Schaefer, H. H., 1999, Topological Vector Spaces, 2nd ed. Springer, New York.

No context found.

H.H. Schaefer and M.P. Wol. Topological vector spaces. Springer, 1999.

No context found.

H.H. Schaefer, Topological Vector Spaces, Macmillan, New-York, 1966.

No context found.

Schaefer, H. H. (1996). Topological vector spaces, Springer Graduate Texts in Mathematics.

No context found.

Schaefer, A., ed.: Topological Vector Spaces. Springer-Verlag (1966)

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H.H. Schaefer, Topological Vector Spaces (Springer, Berlin, 1980).

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H.H. Schaefer, Topological Vector Spaces, 2nd ed. (Springer-Verlag, New York, 1999). 106

No context found.

Schaefer H., "Topological vector spaces," Springer-Verlag, Berlin-HeidelbergNew York, 1971.

No context found.

H.H. Schaefer, Topological Vector Spaces, MacMillan, New York, 1966. 37

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H.H. Schaefer, Topological vector spaces. MacMillan, New York, 1966.

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H. H. Schaefer, Topological Vector Spaces (5th ed.), Grad. Texts in Math., Springer, 1986.

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H. H. Schaefer (1971), Topological Vector Spaces, third printing, Springer Graduate Texts in Math.

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H. H. Schaefer (1971), Topological Vector Spaces, third printing, Springer Graduate Texts in Math.

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H.H. Schaefer (1980). Topological Vector Spaces (4th printing). Springer, New York - Heidelberg - Berlin.

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H.H. Schaefer, Topological Vector Spaces. MacMillan, New York, 1966.

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H. H. SCHAEFER, Topological Vector Spaces, Macmillan, New York, 1966.

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Schaefer, H.H. (1971) Topological Vector Spaces. Springer Verlag, Berlin, Heidelberg and New York.

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