cell division and paradox of Banach – Tarski

I know it's really not formally well stated, but is there a way to formalize the non-equilibrium process of cell division using the Banach-Tarski paradox concept / formalism?

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duality – Nests on Banach spaces and their duels

Let $ X $ to be a Banach space and $ mathcal {E} $ a nest on $ X $.
Take $ f in X ^ {*} $ and suppose:

Is there a function $ h in M ​​^ { bot} $ for some people $ M> N $ so that for all $ epsilon> $ 0, $ | f-h | < epsilon $ ?

I've been thinking about it for a while, but I have not found any evidence of the existence of such a $ h $.

Does anyone know an example of Banach space and a nest where there is not one? $ h $?

Functional Analysis – Is There A Continuous Basic Concept Of A Banach Space?

Yes $ X $ is a Banach space, then a Hamel base $ X $ is a subset $ B $ of $ X $ so that each element of $ X $ can only be written as a linear combination of elements of $ B $. And a Schauder base of $ X $ is a subset $ B $ of $ X $ so that each element of $ X $ can be written only as an infinite linear combination of elements of $ B $?

But my question is: is there a notion of "continuous basis" of a Banach space? This is a subset $ B $ of $ X $ so that each element of $ X $ can be written in terms of a kind of integral involving elements of $ B $.

I do not know what the integral should look like, but one of the possibilities is this. We define a function $ f: mathbb {R} rightarrow X $and we leave $ B $ to be the range of $ f $. And then for everything $ x in X $, there is a unique function $ g: mathbb {R} rightarrow mathbb {R} $ such as $ x = int _ {- infty} ^ infty g (t) f (t) dt $, where it is an integral of Bochner. Does this make sense?

Theory nt.number – Is the upper Banach density always zero compared to a finite subset sequence?

The following question came to me when I read Hindman and Strauss's paper "Density in arbitrary semigroups".

Question: Given an infinite subset $ A $ of $ mathbb {N} $ such as $ A ^ c $ is also infinite, is there a sequence $ mathcal {F} = {F_n } _ {n in mathbb {N}} $ of finite subset $ mathbb {N} $ so the higher density Banach $ A $, that is to say., $ d _ { mathcal {F}} ^ * (A) = 0 ,? $,

or $ d _ { mathcal {F}} ^ * (A): = sup {alpha: forall n in mathbb {N}, exists , n_0 geq n text {and} x in mathbb {N} text {such as} | A cap (F_ {n_0} + x) | geq alpha | F_ {n_0} | } $.

Even if the answer is negative in general, is it true of certain infinite subsets of $ mathbb {N} ,? $

Thank you in advance for your help.

Real Analysis – Operator Regularity of Overlay Generated by Function Between Banach Spaces

Let $ E $, $ F $ to be Banach spaces, $ D $ to be open in $ E $, and $ K =[0,1]$. Given $ varphi colon K times D to F $ J & # 39; calls
$$
varphi ^ sharp colon D ^ K to F ^ K, quad mapsto varphi ( cdot, u ( cdot))
$$

the overlay operator.

I am interested in an overview of how $ varphi ^ sharp $ as a mapping between vector spaces $ V subseteq D ^ K $ and $ W subseteq F ^ K $, both with appropriate standards, inherit regularity $ varphi $.

For example, I can show that $ varphi ^ sharp in C ^ k left (C left (K, D right), C left (K, F right) right)% tag {$ star $} $
provided $ varphi $ and $ partial_2 ^ k varphi $ are continuous.

I am particularly interested in finding $ V $ to be a Hilbert space containing all the piecewise smooth functions K $ to D $.

Thanks in advance!

An example of Banach algebra with a specified property

I asked this question
(Https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions)
but unfortunately nobody answered it. Please, help me find an example of Banach algebra (if any) with the following property:

Non commutative noncommutative Banach algebra $ A $ For who $ aa_0 -a_ {0} a $ is in the annihilator of $ A $ for all $ a in A $.

Right here $ a_0 $ is an element of $ A $ not belonging to his center $ Z (A) $.

Could you please suggest me a good reference (on Banach algebras) with examples like this?

Any help is appreciated.

linear algebra – Dual and bidual on a Banach space

Problem: Let $ E $ to be a normed space on $ mathbb {C} $.

  1. Show that we are able to integrate $ E $ in the second double space (bidual) $ E & # 39; $ of $ E $ by linear isometry, that is to say we can consider $ E $ as a subspace of $ E & # 39; $.
  2. For each subset $ A $ of $ E $, let $ A ^ { perp} = {f in E}: f | _A = 0 } $. Show that a Banach space $ E $ is reflexive $ iff $ for any closed subspace $ F $ of $ E $ we have $ F ^ { perp perp} = J (F) $ in which $ J is canonical entrenchment $ E $ in $ E $.
  3. Fix a continuous $ f: left[ a,b right] rightarrow E $ with $ left[ a,b right] subset mathbb {R} $. Consider $ varphi: E rightarrow mathbb {C} $ given by $ varphi (y): = displaystyle int_a ^ b (y circ f) (t) dt, forall y in E $. CA watch $ varphi in E & # 39; $ and if $ E $ is the Banach space then $ varphi in E $.

My attempt:

  1. Let $ E $ to be a normed space. For each $ x in E $, let $ (Jx) (u) = u (x) $ for everyone $ u in E & # 39; $.

We show that $ J: E rightarrow E & # 39; $ is a linear isometry. Clearly, $ Jx is a linear form on $ E $. Since $ (Jx) (u) = | u (x) | le || u || $ $ || x || $, $ Jx is continuous on $ E $and $ || Jx || le || x || $. So, $ Jx in E & # 39; $. It is common to show that $ J is a linear map of $ E $ in $ E & # 39; $. Take everything $ x in E $. There is $ v in E & # 39; $ such as $ || v || = $ 1 and $ v (x) = || x || $. Therefore
$$ || Jx || = sup {| (Jx) (u) |: || u || le 1 } $$
$$ = sup {| u (x) |: u in E & # 39 ;,| u || le 1 } ge v (x) = || x ||. $$
So $ J is a linear isometry.
Anyone can help me solve the issue number $ 2 $ and $ 3 $? Thank you all!

There are many derivations of discontinuous points on the Banach algebra $ ( mathbb {C ^ {(n)}[0,1], | | _n}) $

This is a 6.2.55 exercise in Garth Dales, Introduction to Banach Algebra

Show that there are many derivatives of discontinuous points on the Banach algebra $ ( mathbb {C ^ {(n)}[0,1], | | _n}) $ or
$$ | f | _n = sum_ {k = 0} ^ {n} frac {1} {k!} | f ^ {(k)} | $$ for everyone $ f in mathbb {C ^ {(n)}[0,1]} $

so if you give reasonable clues, I will be very happy. Thank you

Automatic continuous in Banach algebra

I have the following two questions

1: Let $ A $ and $ B $ to be Banach algebra and suppose that $ B $ is semi-simple. Let $ T: A to B $ to be a homomorphism with $ overline {TA} = B. $ is $ T $ automatically continuously?

2: Let $ A $ and $ B $ to be Banach algebra. Let $ T: A to B $ to be a homomorphism with $ overline {TA} = B. $ is $ T $ automatically continuously?

Any help is very appreciated! Thank you!

Functional Analysis – If the closed unit ball of the Banach space has at least one extreme point, should the Banach space be a double space?

Let $ X $ to be a Banach space.
By the theorems of Banach-Alaoglu and Kerin-Milman, it can be shown that if $ X $ is a double space, so $ X $ must have at least one extreme point.

I'm interested in his conversation.
More precisely,

Question: Let $ X $ to be a Banach space.
If the closed unit ball of $ X $ at least one extreme point, must $ X $ to be a double space?

I think the statement above is negative.
However, I could not produce a counterexample.

In fact, the only Banach spaces that are not double spaces are $ c_0 $ and $ C_0 ( mathbb {R}) $ (The last set is the collection of all real-valued continuous functions that disappear to infinity) because the two sets have no extreme point.