Functor of extension in Hilbert cube and Hilbert space

Abstract

It is shown that if Ω = Q or Ω = 2 , then there exists a functor of extension of maps between Z -sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365-383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics. MSC:57N20, 54C20, 18B30 Keywords

Spaces of measurable functions

“…We conclude from this that such spaces over completely metrizable ones are homeomorphic to Hilbert spaces. In the last part we generalize our results of [15] to nonseparable case. Also the idea of extending maps to AR's via the functors M µ is presented.…”

“…Theorem 2.11 of [5] says that P (X) is homeomorphic to an infinite-dimensional Hilbert space, provided X is completely metrizable and noncompact. Thus it is enough to apply General Scheme of [15] and results of Banakh [2,3] on extending maps and bounded metrics via the functor P .…”

Wearing sensor data integration for promoting the performance skills of music in IoT

“…The nonlinear mapping function 𝜑 that maps the sample space X to the Hilbert space 5  can be expressed as follows.…”

“…The nonlinear mapping function $$$ \varphi $$$ that maps the sample space $$$ \boldsymbol{X} $$$ to the Hilbert space 5 $$$ \mathcal{H} $$$ can be expressed as follows. $$$$ {\displaystyle \begin{array}{cc}\varphi :& \boldsymbol{X}\mathbf{\to}\mathcal{H}\\ {}& x\mathbf{\to}\varphi (x)\end{array}} $$$$ …”