is approximation of arbitrary function spaces
and operators by their analogs in finite dimensions.
Discretization matches the marvelous universal
understanding of computational mathematics as the science of
finite approximations to general (not necessarily metrizable)
compacta. This revolutionary and challenging definition was given in
the joint talk by S. L. Sobolev, L. A. Lyusternik,
and L. V. Kantorovich at the Third All-Union Mathematical Congress
Infinitesimal methods suggest a background, providing new
schemes for discretization of general compact spaces.
As an approximation to a compact space we may take
an arbitrary internal subset containing all standard elements
of the space under approximation.
of the equation
with T:X→Y a bounded linear operator
between some Banach spaces X and Y, consists in choosing
finite-dimensional vector spaces
, the corresponding embeddings
and some operator TN
In this event, the equation
is viewed as a finite-dimensional approximation
to the original problem Tx=y.
in contrast to hypodiscretization
consists in approximating Tx=y by the equation T#
acts between external hyperfinite-dimensional spaces
symbolizes the taking of a nonstandard hull
for spaces and operators.
in the most general sense means reduction to numbers.
Since each number is a measure of quantity,
the idea of scalarization is clearly of a universal importance
to mathematics. The deep roots of scalarization are revealed by
Boolean valued models. Scalarization is effective in operator theory
and multicriteria optimization.
of the ideas of model theory
to analysis projects among the synthetic methods of the present-day
mathematics. This yields new models of numbers,
spaces, and types of equations. The content expands of
all available theorems and algorithms. The whole methodology
of mathematical research is enriched and renewed, opening up
absolutely fantastic opportunities.
We can now use actual infinities and infinitesimals, transform
matrices into numbers, spaces into straight lines, and noncompact spaces into
compact spaces, yet having still uncharted vast territories of new knowledge.
There is no backward traffic in
science, and the new methods
are doomed to reside in the realm of mathematics for ever and in a short
time they will become as elementary and omnipresent in calculuses and
calculations as Banach spaces and linear operators.