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It is proved that the free topological vector space
$\mathbb{V}([0,1])$
contains an isomorphic copy of the free topological vector space
$\mathbb{V}([0,1]^{n})$
for every finite-dimensional cube
$[0,1]^{n}$
, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval
$[0,1]$
to general metrisable spaces. Indeed, we prove that the free topological vector space
$\mathbb{V}(X)$
does not even have a vector subspace isomorphic as a topological vector space to
$\mathbb{V}(X\oplus X)$
, where
$X$
is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.
The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.
We prove that if
$M$
is a
$\text{JBW}^{\ast }$
-triple and not a Cartan factor of rank two, then
$M$
satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of
$M$
onto the unit sphere of another real Banach space
$Y$
extends to a surjective real linear isometry from
$M$
onto
$Y$
.
We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence
$G_{1}\subseteq G_{2}\subseteq \cdots \,$
of topological groups
$G_{n}$
n such that
$G_{n}$
is a subgroup of
$G_{n+1}$
and the latter induces the given topology on
$G_{n}$
, for each
$n\in \mathbb{N}$
. Let
$G$
be the direct limit of the sequence in the category of topological groups. We show that
$G$
induces the given topology on each
$G_{n}$
whenever
$\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$
is an identity neighbourhood in
$G$
for all identity neighbourhoods
$V_{n}\subseteq G_{n}$
. If, moreover, each
$G_{n}$
is complete, then
$G$
is complete. We also show that the weak direct product
$\oplus _{j\in J}G_{j}$
is complete for each family
$(G_{j})_{j\in J}$
of complete Lie groups
$G_{j}$
. As a consequence, every strict direct limit
$G=\cup _{n\in \mathbb{N}}G_{n}$
of finite-dimensional Lie groups is complete, as well as the diffeomorphism group
$\text{Diff}_{c}(M)$
of a paracompact finite-dimensional smooth manifold
$M$
and the test function group
$C_{c}^{k}(M,H)$
, for each
$k\in \mathbb{N}_{0}\cup \{\infty \}$
and complete Lie group
$H$
modelled on a complete locally convex space.
We prove that Krivine’s Function Calculus is compatible with integration. Let
$(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$
be a finite measure space,
$X$
a Banach lattice,
$\mathbf{x}\in X^{n}$
, and
$f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$
a function such that
$f(\cdot ,\unicode[STIX]{x1D714})$
is continuous and positively homogeneous for every
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$
, and
$f(\mathbf{s},\cdot )$
is integrable for every
$\mathbf{s}\in \mathbb{R}^{n}$
. Put
$F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$
and define
$F(\mathbf{x})$
and
$f(\mathbf{x},\unicode[STIX]{x1D714})$
via Krivine’s Function Calculus. We prove that under certain natural assumptions
$F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$
, where the right hand side is a Bochner integral.
The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.
In this paper we use the method of conjugate duality to investigate a class of stochastic optimal control problems where state systems are described by stochastic differential equations with delay. For this, we first analyse a stochastic convex problem with delay and derive the expression for the corresponding dual problem. This enables us to obtain the relationship between the optimalities for the two problems. Then, by linking stochastic optimal control problems with delay with a particular type of stochastic convex problem, the result for the latter leads to sufficient maximum principles for the former.
By homotopy linear algebra we mean the study of linear functors between slices of the ∞-category of ∞-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into ∞-categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality à la Baez, Hoffnung and Walker compatible with this duality. We needed these results to support our work on incidence algebras and Möbius inversion over ∞-groupoids; we hope that they can also be of independent interest.
For a Tychonoff space
$X$
, let
$\mathbb{V}(X)$
be the free topological vector space over
$X$
,
$A(X)$
the free abelian topological group over
$X$
and
$\mathbb{I}$
the unit interval with its usual topology. It is proved here that if
$X$
is a subspace of
$\mathbb{I}$
, then the following are equivalent:
$\mathbb{V}(X)$
can be embedded in
$\mathbb{V}(\mathbb{I})$
as a topological vector subspace;
$A(X)$
can be embedded in
$A(\mathbb{I})$
as a topological subgroup;
$X$
is locally compact.
We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.
We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence
$\ell ^{p}$
spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on
$\ell ^{p}$
’, Sibirsk. Mat. Zh.7 (1966), 146–158].
For a normed infinite-dimensional space, we prove that the family of all locally convex topologies which are compatible with the original norm topology has cardinality greater or equal to
$\mathfrak{c}$
.
This note furnishes an example showing that the main result (Theorem 4) in Toumi [‘When lattice homomorphisms of Archimedean vector lattices are Riesz homomorphisms’, J. Aust. Math. Soc.87 (2009), 263–273] is false.
Metric regularity theory lies in the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. The paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.
The classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.
It is shown that, for any field
$\mathbb{F}\subseteq \mathbb{R}$
, any ordered vector space structure of
$\mathbb{F}^{n}$
with Riesz interpolation is given by an inductive limit of a sequence with finite stages
$(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$
(where
$n$
does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with
$\mathbb{F}$
replaced by the integers,
$\mathbb{Z}$
. Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with
$\mathbb{Q}$
.
It is proved that any surjective morphism
$f:\mathbb{Z}^{{\it\kappa}}\rightarrow K$
onto a locally compact group
$K$
is open for every cardinal
${\it\kappa}$
. This answers a question posed by Hofmann and the second author.
We introduce the concepts of growth and spectral bound for strongly continuous semigroups acting on Fréchet spaces and show that the Banach space inequality s(A) ⩽ ω0(T) extends to the new setting. Via a concrete example of an even uniformly continuous semigroup, we illustrate that for Fréchet spaces effects with respect to these bounds may happen that cannot occur on a Banach space.
The purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.
We give a positive answer to the question of Bouras [‘Almost Dunford–Pettis sets in Banach lattices’, Rend. Circ. Mat. Palermo (2) 62 (2013), 227–236] concerning weak compactness of almost Dunford–Pettis sets in Banach lattices. That is, every almost Dunford–Pettis set in a Banach lattice
$E$
is relatively weakly compact if and only if
$E$
is a
$\mathit{KB}$
-space.