Lemma edit

For all ${\displaystyle x^{\prime \prime }\in B^{\prime \prime },}$  ${\displaystyle \varphi _{1},\ldots ,\varphi _{n}\in X^{\prime }}$  and ${\displaystyle \delta >0,}$  there exists an ${\displaystyle x\in (1+\delta )B}$  such that ${\displaystyle \varphi _{i}(x)=x^{\prime \prime }(\varphi _{i})}$  for all ${\displaystyle 1\leq i\leq n.}$ 

Proof of lemma edit

By the surjectivity of

Now let

Every element of ${\displaystyle z\in (x+Y)\cap (1+\delta )B}$  satisfies ${\displaystyle z\in (1+\delta )B}$  and ${\displaystyle \varphi _{i}(z)=\varphi _{i}(x)=x^{\prime \prime }(\varphi _{i}),}$  so it suffices to show that the intersection is nonempty.

Assume for contradiction that it is empty. Then ${\displaystyle \operatorname {dist} (x,Y)\geq 1+\delta }$  and by the Hahn–Banach theorem there exists a linear form ${\displaystyle \varphi \in X^{\prime }}$  such that ${\displaystyle \varphi {\big \vert }_{Y}=0,\varphi (x)\geq 1+\delta }$  and ${\displaystyle \|\varphi \|_{X^{\prime }}=1.}$  Then ${\displaystyle \varphi \in \operatorname {span} \left\{\varphi _{1},\ldots ,\varphi _{n}\right\}}$ ^[1] and therefore

Proof of theorem edit

Fix ${\displaystyle x^{\prime \prime }\in B^{\prime \prime },}$  ${\displaystyle \varphi _{1},\ldots ,\varphi _{n}\in X^{\prime }}$  and ${\displaystyle \epsilon >0.}$  Examine the set

Let ${\displaystyle J:X\rightarrow X^{\prime \prime }}$  be the embedding defined by ${\displaystyle J(x)={\text{Ev}}_{x},}$  where ${\displaystyle {\text{Ev}}_{x}(\varphi )=\varphi (x)}$  is the evaluation at ${\displaystyle x}$  map. Sets of the form ${\displaystyle U}$  form a base for the weak* topology,^[2] so density follows once it is shown ${\displaystyle J(B)\cap U\neq \varnothing }$  for all such ${\displaystyle U.}$  The lemma above says that for any ${\displaystyle \delta >0}$  there exists a ${\displaystyle x\in (1+\delta )B}$  such that ${\displaystyle x^{\prime \prime }(\varphi _{i})=\varphi _{i}(x),}$  ${\displaystyle 1\leq i\leq n,}$  and in particular ${\displaystyle {\text{Ev}}_{x}\in U.}$  Since ${\displaystyle J(B)\subset B^{\prime \prime },}$  we have ${\displaystyle {\text{Ev}}_{x}\in (1+\delta )J(B)\cap U.}$  We can scale to get ${\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B).}$  The goal is to show that for a sufficiently small ${\displaystyle \delta >0,}$  we have ${\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B)\cap U.}$ 

Directly checking, one has

Note that one can choose ${\displaystyle M}$  sufficiently large so that ${\displaystyle \|\varphi _{i}\|_{X^{\prime }}\leq M}$  for ${\displaystyle 1\leq i\leq n.}$ ^[3] Note as well that ${\displaystyle \|x\|_{X}\leq (1+\delta ).}$  If one chooses ${\displaystyle \delta }$  so that ${\displaystyle \delta M<\epsilon ,}$  then

Hence one gets ${\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B)\cap U}$  as desired.

• Banach–Alaoglu theorem – Theorem in functional analysis
• Bishop–Phelps theorem
• Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
• James' theorem – theorem in mathematicsPages displaying wikidata descriptions as a fallback
• Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space

• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.