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Appendix E - Measure Spaces
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- Functional Analysis
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- 07 July 2022, pp 657-670
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Frontmatter
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- Functional Analysis
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12 - Forms
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- 07 July 2022, pp 409-436
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14 - Trace Class Operators
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- Functional Analysis
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- 07 July 2022, pp 515-550
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Contents
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- 07 July 2022, pp v-viii
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Appendix D - Metric Spaces
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- Functional Analysis
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- 07 July 2022, pp 647-656
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On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field
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- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 3 / August 2022
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- 07 July 2022, pp. 691-704
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Let $\mathcal {N}$
be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$
times weakly locally uniformly differentiable (WLUD$^{k}$
) functions and WLUD$^{\infty }$
functions at a point or on an open subset of $\mathcal {N}$
. Then, we show under what conditions a WLUD$^{\infty }$
function at a point $x_0\in \mathcal {N}$
is analytic in an interval around $x_0$
, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$
and WLUD$^{\infty }$
to functions from $\mathcal {N}^{n}$
to $\mathcal {N}$
, for any $n\in \mathbb {N}$
. Then, we formulate conditions under which a WLUD$^{\infty }$
function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$
is analytic at that point.
1 - Banach Spaces
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- Functional Analysis
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- 07 July 2022, pp 1-34
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9 - The Spectral Theorem for Bounded Normal Operators
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- Functional Analysis
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- 07 July 2022, pp 287-320
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Upper and lower limit oscillation conditions for first-order difference equations
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- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 3 / August 2022
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- 07 July 2022, pp. 618-631
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In this work, we consider the first-order difference equation with general argument
\[ \varDelta x(n)+p(n)x\left( \tau (n)\right) =0,\quad n\geq 0, \]where $(p(n))$
is a sequence of non-negative real numbers, $(\tau (n))$
is a sequence of integers such that $\tau (n)< n$
for$\ n\in \mathbf {N},\,$
and$\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$
Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.
15 - States and Observables
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- 07 July 2022, pp 551-630
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8 - Bounded Operators on Hilbert Spaces
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- 07 July 2022, pp 259-286
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Appendix F - Integration
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- Functional Analysis
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- 07 July 2022, pp 671-680
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4 - Duality
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- Functional Analysis
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- 07 July 2022, pp 117-172
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Appendix A - Zorn’s Lemma
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- 07 July 2022, pp 631-632
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5 - Bounded Operators
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- 07 July 2022, pp 173-210
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3 - Hilbert Spaces
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- Functional Analysis
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- 07 July 2022, pp 89-116
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Appendix G - Notes
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- 07 July 2022, pp 681-700
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Index
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- 07 July 2022, pp 711-722
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Preface
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- 07 July 2022, pp ix-xi
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