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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

2 votes
0 answers
182 views

Example of space which is weak Hahn-Banach smooth but not Hahn-Banach smooth

A Banach space $X$ is said to be Hahn-Banach smooth if every linear functional on $X$ has a unique norm-preserving extension over $X^{**}$. Weak Hahn-Banach smoothness is what if the above condition h …
Tanmoy Paul's user avatar
0 votes
Accepted

Finding the set of best approximation

Similar to $P_Y(x)$, there is no such ready formula for evaluating $P_{B_Y}(x)$, when $Y=\ker (f)$, and so is for $d(x,B_Y)$. In some cases, for instance when $d(x,Y)=d(x,B_Y)$, it is easier to unders …
Tanmoy Paul's user avatar
2 votes
0 answers
168 views

On weak Hahn-Banach smoothness

Let us recall Phelp's property-$U$: A subspace $Y\subset X$ is said to have property-$U$ if every $y^*\in Y^*$ has unique norm preserving extension over $X$. $Y$ is weak Hahn-Banach smooth if $y^*$ ha …
Tanmoy Paul's user avatar
4 votes
1 answer
390 views

Renorming of $C[0,1]$ for a strictly convex dual

Let $C[0,1]$ be the space of all Real valued continuous functions on $[0,1]$ with the usual supremum norm. Does there exist an equivalent renorming on $C[0,1]$ such that the corresponding dual norm is …
Tanmoy Paul's user avatar
1 vote
1 answer
189 views

On examples of subspaces of $C(X)$ for which state spaces are Choquet Simplices

Let $C(X)$ be the Banach space of all Real valued continuous functions on a compact Hausdorff space $X$. What are examples of uniformly closed subspace $\mathcal{A}$ of $C(X)$ such that $\mathcal{A}$ …
Tanmoy Paul's user avatar
6 votes
1 answer
607 views

Whether Krein-Milman property implies Radon-Nikodym property

A Banach space is said to have Krein-Milman property (KMP in short) if every closed bounded convex set of it is a closed convex hull of its extreme points. Eg. Any reflexive space has KMP, $\ell_1$ ha …
Tanmoy Paul's user avatar
4 votes
0 answers
117 views

Korovkin subset of $C(\mathbb{T})$

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_ …
Tanmoy Paul's user avatar
0 votes
Accepted

Peak sets and Choquet boundary of a function algebra

Solution (1). Let $A=\{f\in A(\mathbb{D}):f(0)=f(1)\}$. It follows from the Maximum Modulus Principle that the Choquet boundary $Ch(A)$ does not contain $1$, as $B(A)$ and the set of peak points are …
Tanmoy Paul's user avatar
3 votes
1 answer
348 views

Peak sets and Choquet boundary of a function algebra

I have two problems to ask. Let $A$ be a function algebra of $C(K)$. $t\in K$ is said to be a peak point of $A$ if $\exists~f\in A$ s.t. $|f(t)|=\|f\|$ and $|f(s)|<|f(t)|$ for any $s\neq t$. Suppos …
Tanmoy Paul's user avatar
2 votes
0 answers
323 views

Dual of the space of affine functions

Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\ …
Tanmoy Paul's user avatar
11 votes
1 answer
934 views

Separable bidual but nonseparable third dual

Does there exist a Banach space $X$ such that $X^{**}$ is separable but $X^{***}$ is non-separable? More generally, for every natural $n$ can someone construct an example of Banach space $X$ such tha …
Tanmoy Paul's user avatar
5 votes
1 answer
516 views

Hahn Banach type extension of a Lipschitz map

The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by Kirszb …
Tanmoy Paul's user avatar