[Peterdlax reference for functional analysis exercises] Chapter 3rd Hahn-list Theorem

1. Proof $ (10' $ ).

 

Proof: $ \ rA $: by $ P_k (x) <1 $ Zhi $ \ Bex \ exists \ 0 <A <1, \ ST \ cfrac {x} {A} \ in K. \ EEx $ since $0 $ is an internal point of $ K $, $ \ Bex \ forall \ y, \ exists \ ve = \ ve (y)> 0, \ st | T | <\ cfrac {\ ve} {1-A} \ rA ty \ in K. \ EEx $ is the convex Property of $ K $, $ \ Bex | T | <\ ve \ rA x + ty = A \ cdot \ cfrac {x} {A} + (1-A) \ cdot \ sex {\ cfrac {t} {1-A} y} \ in K. \ EEx $ \ rA $: Set $ x $ to an interior point of $ K $. if $ x = 0 $, $ P_k (x) = 0 $. if $ x \ NEQ 0 $, then $ \ Bex \ exists \ ve = \ ve (x)> 0, \ st | T | <\ ve \ rA x + Tx \ in K. \ EEx $ special, $ \ Bex \ cfrac {x} {\ cfrac {1} {1 + \ cfrac {\ ve} {2 }}= x + \ cfrac {\ ve} {2} X \ in K. \ EEx $ then $ \ Bex P_k (x) \ Leq \ cfrac {1} {1 + \ cfrac {\ ve} {2} <1. \ EEx $

 

2. proof theorem 4.

 

Proof: The proof of (ii) is similar to that of (I ). set $ K =\sed {x \ In X; p (x) <1} $ to $ \ forall \ X, Y \ In K $, $0 <A <1 $, $ \ beex \ Bea P (AX + (1-A) y) & \ Leq P (ax) + P (1-A) y) \\& = ap (x) + (1-A) P (y) \\& <A + (1-A) \\& = A ;\\ AX + (1-A) Y & \ in K. \ EEA \ eeex $ In addition, $0 \ In K $, and for $ \ forall \ Y \ NEQ 0 $, as long as $ \ Bex | T | <\ min \ sed {\ cfrac {1} {| P (y) | + 1 }, \ cfrac {1 }{| P (-y) | + 1 }}, \ EEx $ \ beex \ Bea T> 0 & \ rA P (TY) = T \ cdot P (y) <\ cfrac {P (y) }{| P (y) | + 1} <1, \ t <0 & \ rA P (TY) =-T \ cdot P (-y) <\ cfrac {P (-y) }{| P (-y) | + 1} <1. \ EEA \ eeex $

 

3. Proof: if the condition (17) is changed to $ P ({\ BF a} X) \ Leq p (x) $, theorem 7 remains true.

 

Proof: The proof of theorem 7 is that the conclusion is true.

 

Error message:

 

Page 19, Theorem 5 2nd rows, the number field should be removed.