1,528 research outputs found
A categorical analogue of the monoid semiring construction
This paper introduces and studies a categorical analogue of the familiar
monoid semiring construction. By introducing an axiomatisation of summation
that unifies notions of summation from algebraic program semantics with various
notions of summation from the theory of analysis, we demonstrate that the
monoid semiring construction generalises to cases where both the monoid and the
semiring are categories. This construction has many interesting and natural
categorical properties, and natural computational interpretations.Comment: 34 pages, 5 diagram
Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields
We present an axiomatic approach to finite- and infinite-dimensional
differential calculus over arbitrary infinite fields (and, more generally,
suitable rings). The corresponding basic theory of manifolds and Lie groups is
developed. Special attention is paid to the case of mappings between
topological vector spaces over non-discrete topological fields, in particular
ultrametric fields or the fields of real and complex numbers. In the latter
case, a theory of differentiable mappings between general, not necessarily
locally convex spaces is obtained, which in the locally convex case is
equivalent to Keller's C^k_c-theory.Comment: 70 page
Multi-normed spaces
We modify the very well known theory of normed spaces (E, \norm) within
functional analysis by considering a sequence (\norm_n : n\in\N) of norms,
where \norm_n is defined on the product space for each .
Our theory is analogous to, but distinct from, an existing theory of
`operator spaces'; it is designed to relate to general spaces for , and in particular to -spaces, rather than to -spaces.
After recalling in Chapter 1 some results in functional analysis, especially
in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that
we shall use, we shall present in Chapter 2 our axiomatic definition of a
`multi-normed space' ((E^n, \norm_n) : n\in \N), where (E, \norm) is a
normed space. Several different, equivalent, characterizations of multi-normed
spaces are given, some involving the theory of tensor products; key examples of
multi-norms are the minimum and maximum multi-norm based on a given space.
Multi-norms measure `geometrical features' of normed spaces, in particular by
considering their `rate of growth'. There is a strong connection between
multi-normed spaces and the theory of absolutely summing operators.
A substantial number of examples of multi-norms will be presented.
Following the pattern of standard presentations of the foundations of
functional analysis, we consider generalizations to `multi-topological linear
spaces' through `multi-null sequences', and to `multi-bounded' linear
operators, which are exactly the `multi-continuous' operators. We define a new
Banach space of multi-bounded operators, and show that it
generalizes well-known spaces, especially in the theory of Banach lattices.
We conclude with a theory of `orthogonal decompositions' of a normed space
with respect to a multi-norm, and apply this to construct a `multi-dual' space.Comment: Many update
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