Nondenumerably infinite

It will be noticed that continuous basis sets, with improper Dirao delta functions as scalar products, do not strictly belong to Hilbert space as defined in Section 8.3, where the basis is specifically required by postulate to be denumerably infinite. The nondenumerably infinite sets g> or j actually span what is known as Banach spaces,5 but we shall here conform to the custom among theoretical physicists to oall them Hilbert spaces. 

These expressions are the analogs of Eq. (8-18) defining the matrix of the operators P or Q. The right sides of these equations are matrices only in the sense that has a meaning in Banach space—they have nondenumerable infinite numbers of rows and columns The term matrix is nevertheless a useful one. 

Finally, let the dimension of the space become nondenumerably infinite, turning into a continuum. The sum (4.47) would then be replaced by an integral such as... 

It suffices for most purposes for scientists and engineers to understand that the real numbers, or their geometrical equivalent, the points on a line, are nondenumerably infinite—meaning that they belong to a higher order of infinity than a denumerably infinite set. We, thus, distinguish between variables that have discrete and continuous ranges. A little free hint on... 

But in the usual cases, C represents an infinite set (whether it is denumerable or nondenumerable is immaterial, since at the present in the case of polymer chains it is not enumerable), and (9.14)-(9.16) imply that we would have to perform the usual kind of statistical mechanics (9.1)-(9.5) for an infinite number of subensembles. Then properties are calculated by the ensemble average over the constraints,... 

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