In functional analysis , a branch of mathematics, the Goldstine theorem , named after Herman Goldstine , is stated as follows:
Goldstine theorem. Let X {\displaystyle X} Banach space , then the image of the closed unit ball B ⊆ X {\displaystyle B\subseteq X} B ′ ′ {\displaystyle B^{\prime \prime }} bidual space X ′ ′ {\displaystyle X^{\prime \prime }} weak* -dense subset .The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space of real sequences that converge to zero, c0 space c 0 , {\displaystyle c_{0},} Lp space ℓ ∞ . {\displaystyle \ell ^{\infty }.}
For all x ′ ′ ∈ B ′ ′ , {\displaystyle x^{\prime \prime }\in B^{\prime \prime },} φ 1 , … , φ n ∈ X ′ {\displaystyle \varphi _{1},\ldots ,\varphi _{n}\in X^{\prime }} δ > 0 , {\displaystyle \delta >0,} x ∈ ( 1 + δ ) B {\displaystyle x\in (1+\delta )B} φ i ( x ) = x ′ ′ ( φ i ) {\displaystyle \varphi _{i}(x)=x^{\prime \prime }(\varphi _{i})} 1 ≤ i ≤ n . {\displaystyle 1\leq i\leq n.}
By the surjectivity of { Φ : X → C n , x ↦ ( φ 1 ( x ) , ⋯ , φ n ( x ) ) {\displaystyle {\begin{cases}\Phi :X\to \mathbb {C} ^{n},\\x\mapsto \left(\varphi _{1}(x),\cdots ,\varphi _{n}(x)\right)\end{cases}}} x ∈ X {\displaystyle x\in X} φ i ( x ) = x ′ ′ ( φ i ) {\displaystyle \varphi _{i}(x)=x^{\prime \prime }(\varphi _{i})} 1 ≤ i ≤ n . {\displaystyle 1\leq i\leq n.}
Now let Y := ⋂ i ker φ i = ker Φ . {\displaystyle Y:=\bigcap _{i}\ker \varphi _{i}=\ker \Phi .}
Every element of z ∈ ( x + Y ) ∩ ( 1 + δ ) B {\displaystyle z\in (x+Y)\cap (1+\delta )B} z ∈ ( 1 + δ ) B {\displaystyle z\in (1+\delta )B} φ i ( z ) = φ i ( x ) = x ′ ′ ( φ i ) , {\displaystyle \varphi _{i}(z)=\varphi _{i}(x)=x^{\prime \prime }(\varphi _{i}),}
Assume for contradiction that it is empty. Then dist ( x , Y ) ≥ 1 + δ {\displaystyle \operatorname {dist} (x,Y)\geq 1+\delta } Hahn–Banach theorem there exists a linear form φ ∈ X ′ {\displaystyle \varphi \in X^{\prime }} φ | Y = 0 , φ ( x ) ≥ 1 + δ {\displaystyle \varphi {\big \vert }_{Y}=0,\varphi (x)\geq 1+\delta } ‖ φ ‖ X ′ = 1. {\displaystyle \|\varphi \|_{X^{\prime }}=1.} φ ∈ span { φ 1 , … , φ n } {\displaystyle \varphi \in \operatorname {span} \left\{\varphi _{1},\ldots ,\varphi _{n}\right\}} [ 1] 1 + δ ≤ φ ( x ) = x ′ ′ ( φ ) ≤ ‖ φ ‖ X ′ ‖ x ′ ′ ‖ X ′ ′ ≤ 1 , {\displaystyle 1+\delta \leq \varphi (x)=x^{\prime \prime }(\varphi )\leq \|\varphi \|_{X^{\prime }}\left\|x^{\prime \prime }\right\|_{X^{\prime \prime }}\leq 1,}
Fix x ′ ′ ∈ B ′ ′ , {\displaystyle x^{\prime \prime }\in B^{\prime \prime },} φ 1 , … , φ n ∈ X ′ {\displaystyle \varphi _{1},\ldots ,\varphi _{n}\in X^{\prime }} ϵ > 0. {\displaystyle \epsilon >0.} U := { y ′ ′ ∈ X ′ ′ : | ( x ′ ′ − y ′ ′ ) ( φ i ) | < ϵ , 1 ≤ i ≤ n } . {\displaystyle U:=\left\{y^{\prime \prime }\in X^{\prime \prime }:|(x^{\prime \prime }-y^{\prime \prime })(\varphi _{i})|<\epsilon ,1\leq i\leq n\right\}.}
Let J : X → X ′ ′ {\displaystyle J:X\rightarrow X^{\prime \prime }} J ( x ) = Ev x , {\displaystyle J(x)={\text{Ev}}_{x},} Ev x ( φ ) = φ ( x ) {\displaystyle {\text{Ev}}_{x}(\varphi )=\varphi (x)} x {\displaystyle x} U {\displaystyle U} [ 2] J ( B ) ∩ U ≠ ∅ {\displaystyle J(B)\cap U\neq \varnothing } U . {\displaystyle U.} δ > 0 {\displaystyle \delta >0} x ∈ ( 1 + δ ) B {\displaystyle x\in (1+\delta )B} x ′ ′ ( φ i ) = φ i ( x ) , {\displaystyle x^{\prime \prime }(\varphi _{i})=\varphi _{i}(x),} 1 ≤ i ≤ n , {\displaystyle 1\leq i\leq n,} Ev x ∈ U . {\displaystyle {\text{Ev}}_{x}\in U.} J ( B ) ⊂ B ′ ′ , {\displaystyle J(B)\subset B^{\prime \prime },} Ev x ∈ ( 1 + δ ) J ( B ) ∩ U . {\displaystyle {\text{Ev}}_{x}\in (1+\delta )J(B)\cap U.} 1 1 + δ Ev x ∈ J ( B ) . {\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B).} δ > 0 , {\displaystyle \delta >0,} 1 1 + δ Ev x ∈ J ( B ) ∩ U . {\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B)\cap U.}
Directly checking, one has | [ x ′ ′ − 1 1 + δ Ev x ] ( φ i ) | = | φ i ( x ) − 1 1 + δ φ i ( x ) | = δ 1 + δ | φ i ( x ) | . {\displaystyle \left|\left[x^{\prime \prime }-{\frac {1}{1+\delta }}{\text{Ev}}_{x}\right](\varphi _{i})\right|=\left|\varphi _{i}(x)-{\frac {1}{1+\delta }}\varphi _{i}(x)\right|={\frac {\delta }{1+\delta }}|\varphi _{i}(x)|.}
Note that one can choose M {\displaystyle M} ‖ φ i ‖ X ′ ≤ M {\displaystyle \|\varphi _{i}\|_{X^{\prime }}\leq M} 1 ≤ i ≤ n . {\displaystyle 1\leq i\leq n.} [ 3] ‖ x ‖ X ≤ ( 1 + δ ) . {\displaystyle \|x\|_{X}\leq (1+\delta ).} δ {\displaystyle \delta } δ M < ϵ , {\displaystyle \delta M<\epsilon ,} δ 1 + δ | φ i ( x ) | ≤ δ 1 + δ ‖ φ i ‖ X ′ ‖ x ‖ X ≤ δ ‖ φ i ‖ X ′ ≤ δ M < ϵ . {\displaystyle {\frac {\delta }{1+\delta }}\left|\varphi _{i}(x)\right|\leq {\frac {\delta }{1+\delta }}\|\varphi _{i}\|_{X^{\prime }}\|x\|_{X}\leq \delta \|\varphi _{i}\|_{X^{\prime }}\leq \delta M<\epsilon .}
Hence one gets 1 1 + δ Ev x ∈ J ( B ) ∩ U {\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B)\cap U}
^ Rudin, Walter. Functional Analysis (Second ed.). Lemma 3.9. pp. 63– 64. {{cite book }}: CS1 maint: location (link )^ Rudin, Walter. Functional Analysis (Second ed.). Equation (3) and the remark after. p. 69. {{cite book }}: CS1 maint: location (link )^ Folland, Gerald. Real Analysis: Modern Techniques and Their Applications (Second ed.). Proposition 5.2. pp. 153– 154. {{cite book }}: CS1 maint: location (link )
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