Simple Consequences of Goldstine's Theorem The 2019 Stack Overflow Developer Survey Results...
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Simple Consequences of Goldstine's Theorem
The 2019 Stack Overflow Developer Survey Results Are InConsequence of metrizability proof - disregard, the question is an errorTurning an isometric embedding into a homeomorphismThe the image of the unit ball in X is weak-* dense in the unit ball of X**Unit ball separable $Longrightarrow$ Space separableIf $T:X rightarrow Y$ is a linear operator and $r>0$ such that $r cdot B_Y subseteq T(B_X)$, show $y ||x|| leq M ||y||$.Compactness of the closed unit ball in the weak* topology: the Banach–Alaoglu theoremCan I restrict the condition on the Goldstine's theorem?Metamathematics of the Banach space consequences of the Baire category theoremUsing the Banach-Alaoglu Theorem to show that if $X$ is reflexive then $B_X$ is weakly compact.If a normed space $X$ is reflexive, then $Q(B_X)=B_{X^{**}}$.
$begingroup$
For a normed space $ X $, let $ J : X to X^{**} $ be the natural embedding of $ X $ into $ X^{**} $, and let $ B_X $ and $ B_{X^{**}} $ denote the closed unit balls of $ X $ and $ X^{**} $ respectively.
Goldstine's Theorem: If $ X $ is a normed space then $ J(B_X) $ is weak* dense in $ B_{X^{**}} $.
I want to see the significance of this theorem but I can't find many consequences to this result. Does anyone know of any relatively simple/nice/easy-to-understand consequences of Goldstine's theorem that a beginner in learning functional analysis might be able to understand?
Thanks!
functional-analysis
asked Mar 21 at 4:09
LMWLMW
1257
$endgroup$
add a comment |
$begingroup$
For a normed space $ X $, let $ J : X to X^{**} $ be the natural embedding of $ X $ into $ X^{**} $, and let $ B_X $ and $ B_{X^{**}} $ denote the closed unit balls of $ X $ and $ X^{**} $ respectively.
Goldstine's Theorem: If $ X $ is a normed space then $ J(B_X) $ is weak* dense in $ B_{X^{**}} $.
I want to see the significance of this theorem but I can't find many consequences to this result. Does anyone know of any relatively simple/nice/easy-to-understand consequences of Goldstine's theorem that a beginner in learning functional analysis might be able to understand?
Thanks!
functional-analysis
asked Mar 21 at 4:09
LMWLMW
1257
$endgroup$
add a comment |
$begingroup$
For a normed space $ X $, let $ J : X to X^{**} $ be the natural embedding of $ X $ into $ X^{**} $, and let $ B_X $ and $ B_{X^{**}} $ denote the closed unit balls of $ X $ and $ X^{**} $ respectively.
Goldstine's Theorem: If $ X $ is a normed space then $ J(B_X) $ is weak* dense in $ B_{X^{**}} $.
I want to see the significance of this theorem but I can't find many consequences to this result. Does anyone know of any relatively simple/nice/easy-to-understand consequences of Goldstine's theorem that a beginner in learning functional analysis might be able to understand?
Thanks!
functional-analysis
asked Mar 21 at 4:09
LMWLMW
1257
$endgroup$
For a normed space $ X $, let $ J : X to X^{**} $ be the natural embedding of $ X $ into $ X^{**} $, and let $ B_X $ and $ B_{X^{**}} $ denote the closed unit balls of $ X $ and $ X^{**} $ respectively.
Goldstine's Theorem: If $ X $ is a normed space then $ J(B_X) $ is weak* dense in $ B_{X^{**}} $.
I want to see the significance of this theorem but I can't find many consequences to this result. Does anyone know of any relatively simple/nice/easy-to-understand consequences of Goldstine's theorem that a beginner in learning functional analysis might be able to understand?
Thanks!
functional-analysis
functional-analysis
asked Mar 21 at 4:09
LMWLMW
1257
asked Mar 21 at 4:09
LMWLMW
1257
asked Mar 21 at 4:09
LMWLMW
1257
asked Mar 21 at 4:09
LMWLMW
1257
asked Mar 21 at 4:09
LMWLMW
1257
1257
add a comment |
add a comment |
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