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Recall the Hahn-Banach Theorem in Normed Vector Spaces:

"Let $X$ be a NVS, and let $W$ be a linear subspace of $X$. Then, for any $f_W\in W'$ there exists an extension $f_X\in X'$ such that $\|f_X\|=\|f_W\|$."

And consider its corollary,

"Let $W$ be a proper closed subspace of an NVS $X$ and let $x\in X \setminus W$. Set $\inf_{w\in W}\|x-w\|=\delta\gt0.$ Then, there exists $f\in X'$ such that $\|f\|=1,\,f(w)=0,\, \forall\, w\in W$ and $f(x)=\delta.$"

What does this actually tell us about a bounded linear functional $f\in X'$, and its relation to $W\subset X$? In my notes it says that this corollary is an seperation result, whereas the Hahn-Banach theorem is centered on showing when an extension exists.

I want to say that this corollary is almost the opposite of the Hahn-Banach theorem, since the Hahn-Banach theorem allows us to find an $f_X\in X'$ such that $f_X(w)=f_W(w),\,\forall w \in W$, whereas its corollary states that $f\in X'$ is such that $f(w)=0,\,\forall\, w\in W$. I am looking to get an intuitive understanding of what is going on.

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  • $\begingroup$ Maybe this is too vague, but I think of it as a relationship between the topological structure of a Banach space and its algebraic structure: If a point can be "topologically distinguished" from $W$ (in that it doesn't lie in the closure of $W$), then it can be "algebraically distinguished" from $W$ (by a continuous linear functional). $\endgroup$ – Noah Olander Jan 5 '16 at 19:43

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