# An introduction to real and complex manifolds. by Giuliano Sorani

By Giuliano Sorani

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Example text

E-algebra semi-norm. Let x,y G A, and let e be > 0. We can find s,t € S such that 9(sx) o, . 9{ty) „<,, . E-algebra semi-norm and is 5-multiplicative, we check that s [X + V) 9(st(x + y)) 9{stx) + 9{sty) 9(t)9(sx) + 9(s)9(by) 9(st) ~ 9(st) 9(s)9(t) ^ 0(sx) 9(ty) 9(s) + 9(t) ■ Consequently, we have 9s(x + y) < 9s(x) +9s(y) +2e. Since e is arbitrary, we have proven that 9s(x + y) < 9s(x) + 9s(y). Next, hence 9s(xy) < 6s(x)9s(y). E-algebra semi-norm. Then, it is obviously seen that it is S-multiplicative in the same way as 9.

Such that (x) < ij){x) + e < ||a;|| + e, hence finally (x) < \\x\\. D Definitions and notation: A set F provided with an order relation < is said to be well ordered with respect to the inverse order > if every subset of F admits a maximum element, and then the order > is called a good order. Let > be a good order on the set F and let x € F. If there exists y < x, we call follower of x the element sup{y € F \ y < x}. If there exists z € F such that x is the follower of z, then z is called precedent of x.

M,e) be a neighborhood of ip. ,5i,... ,g'n,e). Each g[ is of the form 4>{fi) (1 < * < " ) , so V(y> o 0, / l t . . ,<£, e)), which finishes proving that (*)" * also is bicontinuous. □ N o t a t i o n : When A' is a dense L-subalgebra of A henceforth, we will consider that Mult(A, \\ . ||) and Mult(A', || . ||) are equal. 9: For every x e A the sequence (|| xn | | " ) n G N has a limit denoted by || x \\Si, satisfying ||x|| S j < ||x|| Vx € A and the mapping f defined in A as f(x) = \\ x ||Sj belongs to SM(A, \\ .