[Peterdlax reference for functional analysis exercises] Chapter 4th application of the Hahn-bananch Theorem

 

 

1. proof: If $ s =\sed {\ mbox {positive integer }$ in section 4.1, $ y $ is the space of the Convergence series, $ \ ell $ is composed of (14) the $ p $ given by (4) is equal to the $ p $ defined by (11.

 

Proof: $ \ Bex p (x) = \ INF _ {x \ Leq Y \ In y} l (y) = \ INF _ {a_n \ Leq B _n, \ sed {B _n} \ In y} \ vlm {n} B _n. \ EEx $ by $ a_n \ Leq B _n $ Zhi $ \ Bex \ VLS {n} a_n \ Leq \ vlm {n} B _n, \ EEx $ and $ \ Bex \ VLS {n} a_n \ Leq p (x ). \ EEx $ on the other hand, for $ \ forall \ ve> 0 $, $ \ forall \ K $, take $ \ Bex y = (A_1, \ cdots, A_k, \ sup _ {n \ geq k} a_n + \ ve, \ cdots, \ sup _ {n \ geq k} a_n + \ ve, \ cdots ), \ EEx $ x \ Leq y $, $ \ Bex p (x) \ Leq L (y) = \ sup _ {n \ geq k} a_n + \ ve. \ EEx $ for $ K $ to obtain the bottom confirmation is $ \ Bex p (x) \ Leq \ VLS {n} a_n + \ ve. \ EEx $ is arbitrary by $ \ ve> 0 $, $ \ Bex p (x) \ Leq \ VLS {n} a_n. \ EEx $

 

2. proof: We can select a list of the bounded series $ \ sed {C_1, C_2, \ cdots} $ that can be added to any Cesaro, both $ \ Bex \ lim _ {n \ To \ infty} C_n = C, \ EEx $ indicates that the arithmetic mean convergence of its parts to $ C $.

 

Proof: Set $ Z $ to the linear space composed of all the bounded real series that Cesaro can add, for $ \ Bex z = (C_1, C_2, \ cdots) \ in Z, \ EEx $ define linear function $ \ Bex L (z) = \ vlm {n} \ cfrac {C_1 + \ cdots + C_n} {n }, \ EEx $ then $ Y \ subset Z $ ($ y $ is the linear space of all converged series) and $ \ bee \ label {4_2_eq} Y \ In Y \ rA L (y) = \ vlm {n} B _n \ quad \ sex {Y = (B _1, B _2, \ cdots )}. \ EEE $ in $ \ beex \ Bea \ cfrac {C_1 + \ cdots + C_n} {n} & =\ cfrac {C_1 + \ cdots + C _ {k-1 }} {n} + \ cfrac {c_k + \ cdots + C_n} {n} \ & \ Leq \ cfrac {C_1 + \ cdots + C _ {k-1} {n} + \ cfrac {n-k + 1} {n} \ sup _ {n \ geq k} C_n \ EEA \ eeex $ order $ n \ To \ infty $ \ Bex L (z) \ Leq \ sup _ {n \ geq k} C_n \ quad \ sex {z = (C_1, C_2, \ cdots )}. \ EEx $ arbitrary by $ K $, $ \ Bex L (z) \ Leq \ INF _ {k \ geq 1} \ sup _ {n \ geq k} C_n = \ VLS {n} C_n. \ EEx $ this indicates that it is in the linear space $ Z $ of $ B $ (linear space composed of all bounded real series, the linear function $ L $ is controlled by $ \ DPS {P (z) =\ VLS {n} C_n} $. according to chapter 7 of theorem 7, $ L $ can be controlled to extend to the whole $ B $. note \ eqref {4_2_eq}. We know $ \ Bex L (x) = \ lim _ {n \ To \ infty} a_n \ quad \ sex {x = (A_1, a_2, \ cdots) \ In B }. \ EEx $

 

3. proof: there is a generalized limit of $ t \ To \ infty $, so that the definition of $ \ sed {T \ In \ BBR; \ t \ geq 0} $ all Bounded Functions on $ x (t) $, the generalized limit satisfies the properties (I) to (iv) in Theorem 3 ).

 

Proof: Set $ x $ to all the Bounded Functions $ x (t) $ defined on $ t \ geq 0 $, $ \ Bex y = \ sed {Y \ In X; \ vlm {t} y (t) \ mbox {exist }}. \ EEx $ definition $ \ Bex L (y) = \ vlm {t} y (t), \ quad Y \ in Y. \ EEx $ is extended from Theorem 1, $ L $ to $ x $. define $ \ Bex \ lim _ {T \ To \ infty} x (t) = L (x), \ quad x \ In X, \ EEx $ this is the generalized limit ($ \ DPS {p (x) =\vls {t} x (t)} $ in the theorem ).

 

 

Error message:

 

Page 28, (20) should be $ C _ {P + \ rock} ={\ bf a }_{-\ rock} C_p $. in fact, $ \ beex \ Bea C _ {P + \ rock} (\ TT) = 1) = {\ bf a }_{-\ ROV} C_p (\ TT ). \ EEA \ eeex $

 

Page 28, note: the three-dimensional sphere should be changed to the sphere in the three-dimensional space.