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''Banach–Mazur operator''

A concept of a computable functional (operator), proposed by S. Banach and S. Mazur [[#References|[1]]] and concerning the computability of a functional (operator) from a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151701.png" /> into a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151702.png" /> as its property to transform any computable sequence of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151703.png" /> into a computable sequence of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151704.png" /> (cf. [[Computable function|Computable function]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151705.png" /> be the set of all one-place general recursive functions (cf. [[General recursive function|General recursive function]]). A functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151706.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151707.png" /> and assuming natural numbers as values is called computable according to Banach–Mazur, or a Banach–Mazur functional, if for any two-place general recursive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151708.png" /> there exists a general recursive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151709.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517010.png" /></td> </tr></table>

(Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517011.png" /> is considered to be a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517012.png" /> for any constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517013.png" />.) All general recursive functionals and everywhere-defined effective functionals (cf. [[Constructive metric space|Constructive metric space]]) are Banach–Mazur functionals. On the other hand, an example of a Banach–Mazur functional which is not identical with any general recursive, and hence neither with any effective, functional is known [[#References|[2]]]. An important property of Banach–Mazur functionals is their continuity [[#References|[1]]]: The values of such a functional on any general recursive function is defined only by a finite number of values of this function.

The concept of computability just outlined can be extended to functions of a real variable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517014.png" /> be the set of computable sequences of computable real numbers; each sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517015.png" /> is defined by two general recursive functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517017.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517019.png" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517020.png" /></td> </tr></table>

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517021.png" /> of a real variable is said to be computable according to Banach–Mazur (the set of such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517022.png" />) if for any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517023.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517024.png" /> the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517025.png" /> also belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517026.png" />. All functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517027.png" /> are continuous at all computable points [[#References|[1]]]. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517028.png" />. The question as to whether all functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517029.png" /> are computably continuous is still (1977) open. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015170/b01517030.png" /> is closed with respect to the sequence of operations used in analysis, so that computable analysis can be successfully developed on this basis [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Mazur,   "Computable analysis" , PWN  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.M. Friedberg,   "4-quantifier completeness: A Banach-Mazur functional not uniformly partial recursive"  ''Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys.'' , '''6''' :  1  (1958)  pp. 1–5</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Markov,   "On constructive functions"  ''Trudy Mat. Inst. Steklov.'' , '''52'''  (1958)  pp. 315–348  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Rogers jr.,   "Theory of recursive functions and effective computability" , McGraw-Hill  (1967)  pp. 164–165</TD></TR></table>



====Comments====
For a more recent exposition of recursive analysis, see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Aberth,   "Computable analysis" , McGraw-Hill  (1980)</TD></TR></table>