Hahn-Banach theorem

The existence of a unique duality mapping could easily be proved. Indeed, let 1 = u G y I u = 1 be a unit sphere in V. According to the Hahn-Banach theorem, for every fixed u G E there exists a unique element u G V such that u = 1, u, u) = 1 due to the strict convexity of V. Let us define... 

Of course, not every measurable set has density. If it is necessary, we can use the Hahn-Banach theorem (Rudin, 1991) and study extensions of p with the following property ... 

Eunctionals p(D) and p(D) are defined for all measurable D. We should stress that such extensions are not unique. Extension of density (24) using the Hahn-Banach theorem for picking up a random integer was used in a very recent work by Adamaszek (2006). 

Naturally, the full proof of the above statements concerning M, the information-minimizing F, and the limit properties, which take us into the general theory of convexity, the Hahn-Banach theorem, and delicate minimization methods, are beyond the scope of this paper, and are being dealt with elsewhere. But once their theoretical justification is established, it is quite striking to see how simple is the formal derivation of an explicit expression of the first right-hand term in Eq. (3). This will be seen later on. 

Feinberg, M., Lavine, R. Thermodynamics based on the Hahn-Banach theorem the Qausius inequality. Arch. Ration. Mech. Anal. 82,203-293 (1983)... 

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