By Lothar Collatz (Auth.)

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**Example text**

If this is substituted into the function Φ, we obtain a new expression Φ = -\\f + h - gf + \\f -h - g\\* + \\f + h\\* -\\f -hf -\\g + h\\* + \\g -h\\\ We now have * = i ( * + Ψ) = έ(ΙΙ* + * +/II 2 + \\g + h -f\\2) - U\\g - h +/II 2 + Wg - h -/ll 2 ) - \\g + h\\2 + \\g - Ä||2. 39), equal to \\g + h II» + H/ll2 and - | | * - A ||2 - ||/|| 2 , respectively; and hence Φ(/, gy h) = 0, that is, (b) holds. In order to prove (a) we investigate the function

(0) - i(|| g II2 - \\g II2 + ill ig II2 - i\\ ig II2) = 0; furthermore,

£)> and also φ(ι) = 0.

E) From pn —► p, pn > ΘΗ there follows: p > Θ ^ . (f) From ΘΗ^Ρη^ση,ση->ΘΗ: Ρη-^ΘΗ. If we use this concept of convergence, several definitions in partially ordered spaces and pseudometric spaces can be given exactly as in topological spaces; for example, a set is closed if the limit element of every convergent sequence belongs to the set. In the special case of a metric space this concept of closedness agrees with the same concept in topological spaces. Lemma* Let ση , τη be the elements of a partially ordered linear space H with ~rn < ση < rn .

R). As a consequence of the third condition for c = — 1 , we conclude II ~ / l l — 11/11· A normed space R is always a (linear) supermetric space, since two elements f,g e R can always be associated, for example, with the real number p(f,g) = | | / — £||, which has all properties of a distance. 31 2 . VARIOUS TYPES OF SPACES Conversely, in a linear, supermetric space R, we may define the distance of / from the null element to be the norm, that is, 11/11 = p(f> ®) provided the homogeneity condition is satisfied.