A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Gateaux and Frechet derivatives on $\mathbb{R}^2$.

I have the following problem: Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by: \begin{equation} f(x,y)= \frac{x^3y}{x^4+y^2}, \quad x \neq 0, y\neq 0 \end{equation} and: \begin{equation} ...
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Example of non-reflexive Banach space

How does one prove that $C^0([0;1],\mathbb{R})$ equipped with the sup norm is not reflexive? I don't understand how to show that the $J$ mapping is not surjective.
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Prove that $(\mathbb R^n,\|\cdot\|_3)^*= (\mathbb R^n,\|\cdot\|_{1.5})$ with full details [on hold]

How can one prove that $(\mathbb{R}^{n},\|\cdot\|_3)^*= (\mathbb{R}^{n},\|\cdot\|_{1.5})$ with full details. How to go about using Hölder's inequality? I know that these are complete inner ...
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145 views

Banach Spaces: Uniform Integral vs. Riemann Integral

Problem Given a finite measure space $\Omega$ and a Banach space $E$. One has strict inclusion: ...
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23 views

Do any $\ell^{p}(\omega)$ have the extension property?

Definition 1. A metric space (normed space) $X$ has the extension property exactly in case, for all finite $A\subseteq X$ and isometric (linear isometric) $f:A\rightarrow X$, there exists an extension ...
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Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...
3
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40 views

Hamel basis and Banach spaces

Suppose $X$ is a linear space and $X$ has a Hamel basis with uncountable number of elements. Does there exist a norm on $X$ such that $X$ is a Banach space with respect to this norm?
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Bijective bounded linear operator is invertible

The following is an exercise from Halmos book "A Hilbert space problem book" : Exercise: If $H$ and $K$ are Hilbert spaces, and if $A$ is a bounded linear transformation that maps $H$ one to one and ...
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Definition of Szlenk index, $w*$ closed set.

I am reading a paper and it has the following: Let $X$ be a separable Banach space. Given $\epsilon>0$, and a $w^*$- closed subset $P$ of $B_{X^*}$, we let $P_\epsilon'=\{x^*\in P \mid $ for all ...
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Converse of uniform boundedness principle

The uniform boundedness principle says if we have a collection of bounded linear operators $\Gamma$ from a banach space $X$ into a normed vector space $Y$, which is pointwise bounded on $X$, i.e. ...
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How to prove that operators are isometry on $\ell^p$/$c$?

Lately I've been studying Banach spaces and isometries, and encountered many explicit isometrys involving $c$, $c_0$, $c^*$, $c_o^*$, $\ell^1$, $\ell^\infty$, etc. (Here $c\subset\ell^\infty$ is the ...
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Verification of a fact, separable Banach spaces, closed subset.

I am reading a proof and it says to verify the following: Suppose $Z$ is a separable Banach space and $F$ is a closed subset of $Z$. Let $\mathcal{O}$ be a countable basis of open subsets of $Z$. We ...
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1answer
33 views

Compact metric implies uncountable w*-dense set

I am reading a proof of the following: Let $X$ be a separable Banach space. The Szlenk index is countable iff $X*$ is separable. In the proof of => it uses the following: If $X^*$ is not separable, ...
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strong epis in the category of banach spaces with linear contractions

In Borceux's Handbook volume 1, page 145, the strong epis in the category of Banach spaces with bounded linear maps of norm <= 1 is characterized as the maps whose restriction on the unit balls is ...
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1answer
39 views

Lp spaces - example of function

Can you give me an example of function which: $$f \in L^{p}[a,b]$$ but $$f \not\in L^{\infty}[a,b]$$ $L^{\infty}[a,b]$ is space of essentially bounded function at interval $[a,b]$ $1 \le p < ...
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1answer
17 views

Prove that $S:F \to \mathcal{L}(E,F)$ is a topological isomorphism

Let $E$ and $F$ be normed spaces, $E \neq \{ 0 \}$ and $x_0 \in E \backslash \{ 0 \}$, $x_0 \in E'$ such that $x_0'(x_0)=1$. Prove that the function $S: F \to \mathcal{L}(E,F)$, $S(y)=T_y$ defined ...
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2answers
30 views

question about the proof that the set C[a,b] with uniform norm is complete

I am trying to understand the proof that the set of continuous function is complete under uniform/supremum norm. First, suppose we have a Cauchy sequence of continuous functions ${f_n(t)}$ with ...
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A question in Banach space

Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively. ...
0
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1answer
18 views

Essential Ideals and Denseness

Let $A$ be a $C^\ast$ algebra. I am wondering if essential ideals in $A$ are dense. It seems like they should, but I don't know how to show it.
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1answer
26 views

Weak convergence + compactness = strong convergence? [duplicate]

Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for ...
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1answer
25 views

Bijection between a Hilbert and a Banach space

I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to ...
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1answer
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Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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Stone-Weierstrass: Summary

This is just a summary. Theorem Given a compact domain $\Omega$. Regard the function space: ...
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Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
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Does every closed, densely operator in a Banach space have an closed, densely defined extension on a Hilbert space?

Assume that $H_1,H_2$ are separable Hilbert spaces, $B$ is a separable Banach space and $H_1\subset B\subset H_2$. Assume further that the inclusion mappings are continuous and have dense images. ...
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1answer
129 views

Comparison between weak convergences in Banach spaces

Let $X$ be a Banach space and let $Y=BC(\mathbb{R},X)$ be the Banach space of all bounded continuous functions from $\mathbb{R}$ to $X$ equipped with the supremum norm. Let $(f_n)_n$ be a sequence of ...
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1answer
235 views

Strongly exposed points/Exposed points

I was studying and I got the next doubt: We suppose that $(X,\|\cdot\|)$ is a Banach space and $C$ it is a convex closed subset of X. We say that $x\in C$ it is an exposed point of $C$ if $\exists ...
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48 views

Are there hypersurfaces with connected complement in a Banach space?

In $\mathbb{R}^n$ it is well-known that a smooth hypersurface $M$ (closed as a subset of $\mathbb{R}^n$) is the zero locus of a global smooth function (whose gradient is nonzero on $M$); from this one ...
2
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1answer
63 views

The Banach–Mazur distance for finite-dimensional $\ell_p$

Let $\ell_p$ denote the usual infinite-dimensional sequence space, and if $n\in\mathbb{Z}^+$ then we let $\ell_p^n$ denote its $n$-dimensional counterpart. Conjecture. Let $1\leq p<\infty$. ...
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Asymptotically nonexpansive mapping that is not nonexpansive

Let $C$ be a nonempty subset of a Banach space $X$. A mapping $T:C\to C$ is said to be asymptotically nonexpansive mappings if for all $n\in \mathbf{N},$ there exists a positive constant $k_n\geq1$ ...
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WOT convergence in the unit ball of B(X)

My questions is (probably) related to: On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$ Does the theorem quoted in the above question, together with ...
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1answer
28 views

Question on banach space over an extension of $\Bbb{Q}_p$

Let $G$ be a compact locally $\Bbb{Q}_p$ analytic group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Let $M$ be a $O[G]$ module. I was reading an article which says : ...
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preserving problem

Define: $$\begin{align*} \varphi &:L^2[0,1] \to L^2[0,1], \\ &(\varphi f)(x)=‎\int_0^x f(t)dt. \end{align*}‎$$ Is it true that, if ‎‎$‎‎B$ is a dense ‎subset ‎of ‎‎$‎‎L^2[0,1]$, then ...
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1answer
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asymptotically nonexpansive mappings

Let $C$ be a nonempty subset of a Banach space $X$. A mapping $T:C\to C$ is said to be asymptotically nonexpansive if for all $n\in \mathbf{N},$ there exists a positive constant $k_n\geq1$ such that ...
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Inverse Function Theorem. On the classical method of proof.

The proof most commonly of the Inverse Function Theorem seen in textbooks of relies on the Banach fixed-point theorem. QUESTION. It is possible to say, using historical references, which ...
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On the cardinality of $\mathbb R \times …\aleph_1 {times}$ and $\mathbb R \times …2^{\aleph_0} \space {times}$

I think I can prove that closure of every countable set in any metric space has cardinality at most $\mathcal c=2^{\aleph _0}$ . So if a metric space is separable i.e. has a countable dense subset $A$ ...
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Completion of a vector space inside a given Banach space

Let $X$ be a normed vector space and $Y$ be a Banach space such that $X$ is continuously embedded into $Y$ (this will be denoted by $X\hookrightarrow Y$ in the sequel). Is it always possible to find ...
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Is the following metric space is a complete metric space?

We have $X=\ell^1$, which contains sequences, which are absolutely convergent, and $d(a_n,b_n) = \sum_{k=1}^{\infty}|a_k-b_k|$. Is this metric space complete or not?
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Scales of Banach Spaces: Literature?

Does someone know a nice reference for: $$E^{-s'}\hookrightarrow E^{-s}\hookrightarrow E^0=E\hookrightarrow E^s\hookrightarrow E^{s'}\quad(s\leq s')$$ (I need a more abstract view; less focus on PDE.) ...
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Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...
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Reference request: linear operators into $L^\infty$ can be extended presrving the norm. [duplicate]

Suppose $X$ is a normed linear space and $Y\subset X$ a linear subspace. I remember that any linear map $L\colon Y\to L^\infty(\Omega)$ can be extended to a linear map $\tilde{L}\colon X\to ...
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Sums of two closed and closed / continuous operators

Let $X$ be a normed space and $A_j:D(A_j)\rightarrow X$ (j=1,2) linear. (i) If $D(A_1)=X$, $A_1$ continuous and A_2 closed. Do we have $A_1+A_2:D(A_1)\cap D(A_2) \rightarrow X$, $x\mapsto A_1x+A_2x$ ...
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Arzela-Ascoli: Proof?

Problem Given a compact domain. Regard the function space: $$\mathcal{C}(\Omega):=\{f:\Omega\to\mathbb{C}:f\text{ continuous}\}$$ Consider a bounded family: ...
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1answer
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inequality in a banach space

Let $(X,\left\| \cdot \right\|)$ be a Banach space. For each $i=1,\cdots,n$, let $a_i\in X$ and $\alpha_i\in\mathbb{R}$. Suppose that $0\leq\alpha_i\leq M$ for all $i=1,\cdots,n$. Question: Is it ...
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Quotient Rings of Algebras

So let us take the commutative Banach algebra $B=\mathscr{l}^1(\mathbb{Z}_n)$ over $\mathbb{R}$ with convolution as multiplication $(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ...
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Does a Banach space always contain an element of arbitrarily large norm?

Let $X$ be a Banach space. Or say, even just a normed linear space. Let $N \in \mathbb N$. Does there exist $x \in X $ with $\|x\| \ge N$? If $X$ is a Banach space then its unit ball is also a ...
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A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the ...
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2answers
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If $E\subset X^{*}$ is bounded, then so is its weak* closure

If $X$ is a Banach space and $E\subset X^{*}$ is norm-bounded, I've shown that its weak* closure is also norm-bounded using Alaoglu's theorem. But perhaps using Alaoglu's theorem is not necessary? ...
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1answer
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Show that $X$ is Banach space and describe $X^*$.

Let $X=L^2(\mu)\times L^2(\mu)=\{(f,g)|f,g\in L^2(\mu)\}$ be the linear space normed by $\|(f,g)\|=(\|f\|_2^3+\|g\|_2^3)^{1/3}$. Show that $X$ is Banach space and describe $X^*$. My Work: We ...
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Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...