The impact of organ motion and the appliance of mitigation strategies on the effectiveness of hypoxia-guided proton therapy for non-small cell lung cancer.

BACKGROUND AND PURPOSE To investigate the impact of organ motion on hypoxia-guided proton therapy treatments for non-small cell lung cancer (NSCLC) patients. MATERIALS AND METHODS Hypoxia PET and 4D imaging data of six NSCLC patients were used to simulate hypoxia-guided proton therapy with different motion mitigation strategies including rescanning, breath-hold, respiratory gating and tumour tracking. Motion-induced dose degradation was estimated for treatment plans with dose painting of hypoxic tumour sub-volumes at escalated dose levels. Tumour control probability (TCP) and dosimetry indices were assessed to weigh the clinical benefit of dose escalation and motion mitigation. In addition, the difference in normal tissue complication probability (NTCP) between escalated proton and photon VMAT treatments have been assessed. RESULTS Motion-induced dose degradation was found for target coverage (CTV V95% up to -4%) and quality of the dose-escalation-by-contour (QRMS up to 6%) as a function of motion amplitude and amount of dose escalation. The TCP benefit coming from dose escalation (+4-13%) outweighs the motion-induced losses (<2%). Significant average NTCP reductions of dose-escalated proton plans were found for lungs (-14%), oesophagus (-10%) and heart (-16%) compared to conventional VMAT plans. The best plan dosimetry was obtained with breath hold and respiratory gating with rescanning. CONCLUSION NSCLC affected by hypoxia appears to be a prime target for proton therapy which, by dose-escalation, allows to mitigate hypoxia-induced radio-resistance despite the sensitivity to organ motion. Furthermore, substantial reduction in normal tissue toxicity can be expected compared to conventional VMAT. Accessibility and standardization of hypoxia imaging and clinical trials are necessary to confirm these findings in a clinical setting

Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

Duals of variable exponent Hörmander spaces (0<p−≤p+≤10< p^- \le p^+ \le 1) and some applications

In this paper we characterize the dual \bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' of the variable exponent H\"or\-man\-der space \B^c_{p(\cdot)} (\Omega) when the exponent $p(\cdot)$ satisfies the conditions $0 < p^- \le p^+ \le 1$, the Hardy-Littlewood maximal operator $M$ is bounded on $L_{p(\cdot)/p_0}$ for some $0 < p_0 < p^-$ and $\Omega$ is an open set in $\R^n$. It is shown that the dual \bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' is isomorphic to the H\"ormander space \B^{\mathrm{loc}}_\infty (\Omega) (this is the $p^+ \le 1$ counterpart of the isomorphism \bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' \simeq \B^{\mathrm{loc}}_{\widetilde{p'(\cdot)}} (\Omega), $1 < p^- \le p^+ < \infty$, recently proved by the authors) and hence the representation theorem \bigl( \B^c_{p(\cdot)} (\Omega) \bigr)' \simeq l^{\N}_\infty is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of \B^c_{p(\cdot)} (\Omega) and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for $p(\cdot) \equiv p$, $0 < p < 1$, are also given (e.g. \B^c_p (\Omega) does not contain any infinite-dimensional $q$-Banach subspace with $p < q \le 1$ or the quasi-Banach space \B_p \cap \E'(Q) contains a copy of $l_p$ when $Q$ is a cube in $\R^n$). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent H\"ormander spaces is proposed.J. Motos is partially supported by grant MTM2011-23164 from the Spanish Ministry of Science and Innovation. The authors wish to thank the referees for the careful reading of the manuscript and for many helpful suggestions and remarks that improved the exposition. In particular, the remark immediately following Theorem 2.1 and the Question 2 were motivated by the comments of one of them.Motos Izquierdo, J.; Planells Gilabert, MJ.; Talavera Usano, CF. (2015). Duals of variable exponent Hörmander spaces ($0< p^- \le p^+ \le 1$) and some applications. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 109(2):657-668. https://doi.org/10.1007/s13398-014-0209-zS6576681092Aboulaich, R., Meskine, D., Souissi, A.: New diffussion models in image processing. Comput. Math. Appl. 56(4), 874–882 (2008)Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164(3), 213–259 (2002)Bastero, J.: $$l^q$$ l q -subspaces of stable $$p$$ p -Banach spaces, $$0 < p \le 1$$ 0 < p ≤ 1 . Arch. Math. (Basel) 40, 538–544 (1983)Boas, R.P.: Entire functions. Academic Press, London (1954)Boza, S.: Espacios de Hardy discretos y acotación de operadores. Dissertation, Universitat de Barcelona (1998)Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue spaces, foundations and harmonic analysis. Birkhäuser, Basel (2013)Cruz-Uribe, D.: SFO, A. Fiorenza, J. M. Martell, C. Pérez: The boundedness of classical operators on variable $$L^p$$ L p spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and sobolev spaces with variable exponents. lecture notes in mathematics, vol. 2007. Springer, Berlin, Heidelberg (2011)Hörmander, L.: The analysis of linear partial operators II, Grundlehren 257. Springer, Berlin, Heidelberg (1983)Hörmander, L.: The analysis of linear partial operators I, Grundlehren 256. Springer, Berlin, Heidelberg (1983)Kalton, N.J., Peck, N.T., Roberts, J.W.: An $$F$$ F -space sampler, London Mathematical Society Lecture Notes, vol. 89. Cambridge University Press, Cambridge (1985)Kalton, N.J.: Banach envelopes of non-locally convex spaces. Canad. J. Math. 38(1), 65–86 (1986)Kalton, N.J., Mitrea, M.: Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Am. Math. Soc. 350(10), 3903–3922 (1998)Kalton, N.J.: Quasi-Banach spaces, Handbook of the Geometry of Banach Spaces, vol. 2. In: Johnson, W.B., Lindenstrauss, J. (eds.), pp. 1099–1130. Elsevier, Amsterdam (2003)Köthe, G.: Topological vector spaces I. Springer, Berlin, Heidelberg (1969)Motos, J., Planells, M.J., Talavera, C.F.: On variable exponent Lebesgue spaces of entire analytic functions. J. Math. Anal. Appl. 388, 775–787 (2012)Motos, J., Planells, M.J., Talavera, C.F.: A note on variable exponent Hörmander spaces. Mediterr. J. Math. 10, 1419–1434 (2013)Stiles, W.J.: Some properties of $$l_p$$ l p , $$0 < p < 1$$ 0 < p < 1 . Studia Math. 42, 109–119 (1972)Triebel, H.: Theory of function spaces. Birkhäuser, Basel (1983)Vogt, D.: Sequence space representations of spaces of test functions and distributions. In: Zapata, G.I. (ed.) Functional analysis, holomorphy and approximation theory, Lecture Notes in Pure and Applied Mathematics, vol. 83, pp. 405–443 (1983

Algebraic entropy in locally linearly compact vector spaces

We introduce algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as a natural extension of the algebraic entropy for endomorphisms of discrete vector spaces studied in Giordano Bruno and Salce (Arab J Math 1:69\u201387, 2012). We show that the main properties continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem

Twisted convolution and Moyal star product of generalized functions

We consider nuclear function spaces on which the Weyl-Heisenberg group acts continuously and study the basic properties of the twisted convolution product of the functions with the dual space elements. The final theorem characterizes the corresponding algebra of convolution multipliers and shows that it contains all sufficiently rapidly decreasing functionals in the dual space. Consequently, we obtain a general description of the Moyal multiplier algebra of the Fourier-transformed space. The results extend the Weyl symbol calculus beyond the traditional framework of tempered distributions.Comment: LaTeX, 16 pages, no figure

Improving insect conservation management through insect monitoring and stakeholder involvement

In recent years, the decline of insect biodiversity and the imminent loss of provided ecosystem functions and services has received public attention and raised the demand for political action. The complex, multi-causal contributors to insect decline require a broad interdisciplinary and cross-sectoral approach that addresses ecological and social aspects to find sustainable solutions. The project Diversity of Insects in Nature protected Areas (DINA) assesses insect communities in 21 nature reserves in Germany, and considers interactions with plant diversity, pesticide exposure, spatial and climatic factors. The nature reserves border on agricultural land, to investigate impacts on insect diversity. Part of the project is to obtain scientific data from Malaise traps and their surroundings, while another part involves relevant stakeholders to identify opportunities and obstacles to insect diversity conservation. Our results indicate a positive association between insect richness and biomass. Insect richness was negatively related to the number of stationary pesticides (soil and vegetation), pesticides measured in ethanol, the amount of area in agricultural production, and precipitation. Our qualitative survey along with stakeholder interviews show that there is general support for insect conservation, while at the same time the stakeholders expressed the need for more information and data on insect biodiversity, as well as flexible policy options. We conclude that conservation management for insects in protected areas should consider a wider landscape. Local targets of conservation management will have to integrate different stakeholder perspectives. Scientifically informed stakeholder dialogues can mediate conflicts of interests, knowledge, and values to develop mutual conservation scenarios

New Protocetid Whale from the Middle Eocene of Pakistan: Birth on Land, Precocial Development, and Sexual Dimorphism

BACKGROUND: Protocetidae are middle Eocene (49-37 Ma) archaeocete predators ancestral to later whales. They are found in marine sedimentary rocks, but retain four legs and were not yet fully aquatic. Protocetids have been interpreted as amphibious, feeding in the sea but returning to land to rest. METHODOLOGY/PRINCIPAL FINDINGS: Two adult skeletons of a new 2.6 meter long protocetid, Maiacetus inuus, are described from the early middle Eocene Habib Rahi Formation of Pakistan. M. inuus differs from contemporary archaic whales in having a fused mandibular symphysis, distinctive astragalus bones in the ankle, and a less hind-limb dominated postcranial skeleton. One adult skeleton is female and bears the skull and partial skeleton of a single large near-term fetus. The fetal skeleton is positioned for head-first delivery, which typifies land mammals but not extant whales, evidence that birth took place on land. The fetal skeleton has permanent first molars well mineralized, which indicates precocial development at birth. Precocial development, with attendant size and mobility, were as critical for survival of a neonate at the land-sea interface in the Eocene as they are today. The second adult skeleton is the most complete known for a protocetid. The vertebral column, preserved in articulation, has 7 cervicals, 13 thoracics, 6 lumbars, 4 sacrals, and 21 caudals. All four limbs are preserved with hands and feet. This adult is 12% larger in linear dimensions than the female skeleton, on average, has canine teeth that are 20% larger, and is interpreted as male. Moderate sexual dimorphism indicates limited male-male competition during breeding, which in turn suggests little aggregation of food or shelter in the environment inhabited by protocetids. CONCLUSIONS/SIGNIFICANCE: Discovery of a near-term fetus positioned for head-first delivery provides important evidence that early protocetid whales gave birth on land. This is consistent with skeletal morphology enabling Maiacetus to support its weight on land and corroborates previous ideas that protocetids were amphibious. Specimens this complete are virtual 'Rosetta stones' providing insight into functional capabilities and life history of extinct animals that cannot be gained any other way

Bounded and compact multipliers between Bergman and Hardy spaces

This paper studies the boundedness and compactness of the coefficient multiplier operators between various Bergman spaces A p and Hardy spaces H q . Some new characterizations of the multipliers between the spaces with exponents 1 or 2 are derived which, in particular, imply a Bergman space analogue of the Paley-Rudin Theorem on sparse sequences. Hardy and Bergman spaces are shown to be linked using mixed-norm spaces, and this linkage is used to improve a known result on ( A p , A 2 ), 1< p <2.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42949/1/20_2005_Article_BF01225524.pd

On realcompact topological vector spaces

[EN] This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Frechet spaces, (L F)-spaces, and their duals, (D F)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (L F)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no. NN201 2740 33 and for the both authors by the project MTM2008-01502 of the Spanish Ministry of Science and Innovation.Kakol, JM.; López Pellicer, M. (2011). On realcompact topological vector spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 105(1):39-70. https://doi.org/10.1007/s13398-011-0003-0S39701051Argyros S., Mercourakis S.: On weakly Lindelöf Banach spaces. Rocky Mountain J. Math. 23(2), 395–446 (1993). doi: 10.1216/rmjm/1181072569Arkhangel’skii, A. V.: Topological Function Spaces, Mathematics and its Applications, vol. 78, Kluwer, Dordrecht (1992)Batt J., Hiermeyer W.: On compactness in L p (μ, X) in the weak topology and in the topology σ(L p (μ, X), L p (μ,X′)). Math. Z. 182, 409–423 (1983)Baumgartner J.E., van Douwen E.K.: Strong realcompactness and weakly measurable cardinals. Topol. Appl. 35, 239–251 (1990). doi: 10.1016/0166-8641(90)90109-FBierstedt K.D., Bonet J.: Stefan Heinrich’s density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces. Math. Nachr. 35, 149–180 (1988)Cascales B.: On K-analytic locally convex spaces. Arch. Math. 49, 232–244 (1987)Cascales B., Ka̧kol J., Saxon S.A.: Weight of precompact subsets and tightness. J. Math. Anal. Appl. 269, 500–518 (2002). doi: 10.1016/S0022-247X(02)00032-XCascales B., Ka̧kol J., Saxon S.A.: Metrizability vs. Fréchet–Urysohn property. Proc. Am. Math. Soc. 131, 3623–3631 (2003)Cascales B., Namioka I., Orihuela J.: The Lindelöf property in Banach spaces. Stud. Math. 154, 165–192 (2003). doi: 10.4064/sm154-2-4Cascales B., Oncina L.: Compactoid filters and USCO maps. J. Math. Anal. Appl. 282, 826–843 (2003). doi: 10.1016/S0022-247X(03)00280-4Cascales B., Orihuela J.: On compactness in locally convex spaces, Math. Z. 195(3), 365–381 (1987). doi: 10.1007/BF01161762Cascales B., Orihuela J.: On pointwise and weak compactness in spaces of continuous functions. Bull. Soc. Math. Belg. Ser. B 40(2), 331–352 (1988) Journal continued as Bull. Belg. Math. Soc. Simon StevinDiestel J.: $${L^{1}_{X}}$$ is weakly compactly generated if X is. Proc. Am. Math. Soc. 48(2), 508–510 (1975). doi: 10.2307/2040292van Douwen E.K.: Prime mappings, number of factors and binary operations. Dissertationes Math. (Rozprawy Mat.) 199, 35 (1981)Drewnowski L.: Resolutions of topological linear spaces and continuity of linear maps. J. Math. Anal. Appl. 335(2), 1177–1195 (2007). doi: 10.1016/j.jmaa.2007.02.032Engelking R.: General Topology. Heldermann Verlag, Lemgo (1989)Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. Canadian Mathematical Society. Springer, Berlin (2001)Ferrando J.C.: A weakly analytic space which is not K-analytic. Bull. Aust. Math. Soc. 79(1), 31–35 (2009). doi: 10.1017/S0004972708000968Ferrando J.C.: Some characterization for υ X to be Lindelöf Σ or K-analytic in term of C p (X). Topol. Appl. 156(4), 823–830 (2009). doi: 10.1016/j.topol.2008.10.016Ferrando J.C., Ka̧kol J.: A note on spaces C p (X) K-analytic-framed in $${\mathbb{R}^{X} }$$ . Bull. Aust. Math. Soc. 78, 141–146 (2008)Ferrando J.C., Ka̧kol J., López-Pellicer M.: Bounded tightness conditions and spaces C(X). J. Math. Anal. Appl. 297, 518–526 (2004)Ferrando J.C., Ka̧kol J., López-Pellicer M.: A characterization of trans-separable spaces. Bull. Belg. Math. Soc. Simon Stevin 14, 493–498 (2007)Ferrando, J.C., Ka̧kol, J., López-Pellicer, M.: Metrizability of precompact sets: an elementary proof. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat. RACSAM 99(2), 135–142 (2005). http://www.rac.es/ficheros/doc/00173.pdfFerrando J.C., Ka̧kol J., López-Pellicer M., Saxon S.A.: Tightness and distinguished Fréchet spaces. J. Math. Anal. Appl. 324, 862–881 (2006). doi: 10.1016/j.jmaa.2005.12.059Ferrando J.C., Ka̧kol J., López-Pellicer M., Saxon S.A.: Quasi-Suslin weak duals. J. Math. Anal. Appl. 339(2), 1253–1263 (2008). doi: 10.1016/j.jmaa.2007.07.081Floret, K.: Weakly compact sets. Lecture Notes in Mathematics, vol. 801, Springer, Berlin (1980)Gillman L., Henriksen M.: Rings of continuous functions in which every finitely generated ideal is principial. Trans. Am. Math. Soc. 82, 366–391 (1956). doi: 10.2307/1993054Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand Reinhold Company, New York (1960)Grothendieck A.: Sur les applications linéaires faiblement compactes d’espaces du type C(K). Can. J. Math. 5, 129–173 (1953)Gullick D., Schmets J.: Separability and semi-norm separability for spaces of bounded continuous functions. Bull. R. Sci. Lige 41, 254–260 (1972)Hager A.W.: Some nearly fine uniform spaces. Proc. Lond. Math. Soc. 28, 517–546 (1974). doi: 10.1112/plms/s3-28.3.517Howes N.R.: On completeness. Pacific J. Math. 38, 431–440 (1971)Isbell, J.R.: Uniform spaces. In: Mathematical Surveys 12, American Mathematical Society, Providence (1964)Ka̧kol J., López-Pellicer M.: Compact coverings for Baire locally convex spaces. J. Math. Anal. Appl. 332, 965–974 (2007). doi: 10.1016/j.jmaa.2006.10.045Ka̧kol, J., López-Pellicer, M.: A characterization of Lindelöf Σ-spaces υ X (preprint)Ka̧kol J., López-Pellicer M., Śliwa W.: Weakly K-analytic spaces and the three-space property for analyticity. J. Math. Anal. Appl. 362(1), 90–99 (2010). doi: 10.1016/j.jmaa.2009.09.026Ka̧kol J., Saxon S.: Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex topology. J. Lond. Math. Soc. 66(2), 388–406 (2002)Ka̧kol J., Saxon S., Todd A.T.: Pseudocompact spaces X and df-spaces C c (X). Proc. Am. Math. Soc. 132, 1703–1712 (2004)Ka̧kol J., Śliwa W.: Strongly Hewitt spaces. Topology Appl. 119(2), 219–227 (2002). doi: 10.1016/S0166-8641(01)00063-3Khan L.A.: Trans-separability in spaces of continuous vector-valued functions. Demonstr. Math. 37, 61–67 (2004)Khan L.A.: Trans-separability in the strict and compact-open topologies. Bull. Korean Math. Soc. 45, 681–687 (2008). doi: 10.4134/BKMS.2008.45.4.681Khurana S.S.: Weakly compactly generated Fréchet spaces. Int. J. Math. Math. Sci. 2(4), 721–724 (1979). doi: 10.1155/S0161171279000557Kirk R.B.: A note on the Mackey topology for (C b (X)*,C b (X)). Pacific J. Math. 45(2), 543–554 (1973)Köthe G.: Topological Vector Spaces I. Springer, Berlin (1969)Kubiś W., Okunev O., Szeptycki P.J.: On some classes of Lindelöf Σ-spaces. Topol. Appl. 153(14), 2574–2590 (2006). doi: 10.1016/j.topol.2005.09.009Künzi H.P.A., Mršević M., Reilly I.L., Vamanamurthy M.K.: Pre-Lindelöf quasi-pseudo-metric and quasi-uniform spaces. Mat. Vesnik 46, 81–87 (1994)Megginson R.: An Introduction to Banach Space Theory. Springer, Berlin (1988)Michael E.: ℵ0-spaces. J. Math. Mech. 15, 983–1002 (1966)Nagami K.: Σ-spaces. Fund. Math. 61, 169–192 (1969)Narayanaswami P.P., Saxon S.A.: (LF)-spaces, quasi-Baire spaces and the strongest locally convex topology. Math. Ann. 274, 627–641 (1986). doi: 10.1007/BF01458598Negrepontis S.: Absolute Baire sets. Proc. Am. Math. Soc. 18(4), 691–694 (1967). doi: 10.2307/2035440Orihuela J.: Pointwise compactness in spaces of continuous functions. J. Lond. Math. Soc. 36(2), 143–152 (1987). doi: 10.1112/jlms/s2-36.1.143Orihuela, J.: On weakly Lindelöf Banach spaces. In: Bierstedt, K.D. et al. (eds.) Progress in Functional Analysis, pp. 279–291. Elsvier, Amsterdam (1992). doi: 10.1016/S0304-0208(08)70326-8Orihuela J., Schachermayer W., Valdivia M.: Every Readom–Nikodym Corson compact space is Eberlein compact. Stud. Math. 98, 157–174 (1992)Orihuela, J., Valdivia, M.: Projective generators and resolutions of identity in Banach spaces. Rev. Mat. Complut. 2(Supplementary Issue), 179–199 (1989)Pérez Carreras P., Bonet J.: Barrelled Locally Convex Spaces, Mathematics Studies 131. North-Holland, Amsterdam (1987)Pfister H.H.: Bemerkungen zum Satz über die separabilität der Fréchet-Montel Raüme. Arch. Math. (Basel) 27, 86–92 (1976). doi: 10.1007/BF01224645Robertson N.: The metrisability of precompact sets. Bull. Aust. Math. Soc. 43(1), 131–135 (1991). doi: 10.1017/S0004972700028847Rogers C.A., Jayne J.E., Dellacherie C., Topsøe F., Hoffman-Jørgensen J., Martin D.A., Kechris A.S., Stone A.H.: Analytic Sets. Academic Press, London (1980)Saxon S.A.: Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology. Math. Ann. 197(2), 87–106 (1972). doi: 10.1007/BF01419586Schawartz L.: Radom Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press, Oxford (1973)Schlüchtermann G., Wheller R.F.: On strongly WCG Banach spaces. Math. Z. 199(3), 387–398 (1988). doi: 10.1007/BF01159786Schlüchtermann G., Wheller R.F.: The Mackey dual of a Banach space. Note Math. 11, 273–287 (1991)Schmets, J.: Espaces de functions continues. Lecture Notes in Mathematics, vol 519, Springer-Verlag, Berlin-New York (1976)Talagrand M.: Sur une conjecture de H. H. Corson. Bull. Soc. Math. 99, 211–212 (1975)Talagrand M.: Espaces de Banach faiblement K-analytiques. Ann. Math. 110, 407–438 (1979)Talagrand M.: Weak Cauchy sequences in L 1(E). Am. J. Math. 106(3), 703–724 (1984). doi: 10.2307/2374292Tkachuk V.V.: A space C p (X) is dominated by irrationals if and only if it is K-analytic. Acta Math. Hungar. 107(4), 253–265 (2005)Tkachuk V.V.: Lindelöf Σ-spaces: an omnipresent class. RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat. 104(2), 221–244 (2010). doi: 10.5052/RACSAM.2010.15Todd A.R., Render H.: Continuous function spaces, (db)-spaces and strongly Hewitt spaces. Topol. Appl. 141, 171–186 (2004). doi: 10.1016/j.topol.2003.12.005Valdivia M.: Topics in Locally Convex Spaces, Mathematics Studies 67. North-Holland, Amsterdam (1982)Valdivia M.: Espacios de Fréchet de generación débilmente compacta. Collect. Math. 38, 17–25 (1987)Valdivia M.: Resolutions of identity in certain Banach spaces. Collect. Math. 38, 124–140 (1988)Valdivia M.: Resolutions of identity in certain metrizable locally convex spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.) 83, 75–96 (1989)Valdivia M.: Projective resolutions of identity in C(K) spaces. Arch. Math. (Basel) 54, 493–498 (1990)Valdivia, M.: Resoluciones proyectivas del operador identidad y bases de Markusevich en ciertos espacios de Banach. Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.) 84, 23–34Valdivia M.: Quasi-LB-spaces. J. Lond. Math. Soc. 35(2), 149–168 (1987). doi: 10.1112/jlms/s2-35.1.149Walker, R.C.: The Stone-Čech compactification Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 83. Springer, Berlin (1974