## 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 $$K to D$$.

## An example of Banach algebra with a specified property

(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.

## Real Analysis – The Neat Banach Space Concept

In this article, the authors used the notion of ordered Banach space.

Definition: Let $$mathcal E$$ to be a Banach space with the standard $$left | . right |$$, whose positive cone is defined by $$K = {x in mathcal E : : x geq 0 }$$. then $$( mathcal E, left |. right |)$$ is now a partially ordered Banach space with the order relationship $$sqsubseteq$$ induced by the cone $$K$$.

and they used this result:

Theorem:Let $$( mathcal E, left |. right |, sqsubseteq)$$ to be an orderly space of Banach, whose positive cone $$K$$ Is normal(?). Let $$(u_n) _ {n in mathbb N}$$ a monotonous sequence (let's say $$u_n sqsubseteq u_ {n + 1}$$), such as $${u_n}$$ has a convergent subsequence. Then the whole sequence $$u_n$$ is convergent?

I have two questions here:

$$1.$$ What is it? $$a sqsubseteq u _ {n + 1}$$ means, is there an interpretation of that?

$$2.$$ I wonder why the whole sequence $$u_n$$ is convergent?