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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When is a norm of identity one?

Is there a specific condition that makes a norm (any norm) of identity equal to one in any Banach spaces? Thanks.
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Projection on closed subspace of $L^1$, $L^{\infty}$

For $p=1,\infty$ let $K$ be a closed subspace of $L^p(\mathbb{R},m)$. According to this question, it should be easy to find examples of $K$ and $f\in L^p(\mathbb{R},m)$ such that there exists a ...
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38 views

Projection on closed subspace of $L^p$, $1<p<\infty$

Let $1<p<\infty$ and $K$ be a closed subspace of $L^p(X, \mathcal{M}, \mu)$. If $f\in L^p$ then there exists a unique $h\in K$ such that $||f-h||_p$ equals $$ \text{dist}(f,K)=\inf_{g\in ...
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Closed hyperplanes of a Banach space are isomorphic

Let $X$ be a Banach space and $S_1$ and $S_2$ two of its closed subspaces of co-dimension 1. Prove that they are isomorphic (i.e. there is a bounded bijective linear operator between them).
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32 views

Is this a Hilbert space?

For $n\geq 2$, we let $\mathcal{H}$ be the complex vector space of all complex-valued functions on $[0,1]$ such that (a) $f(0)=0$, (b) for $1\leq k\leq n-1$, $f^{(k)}$ exists everywhere and is ...
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1answer
48 views

Can the ball $B(0,r_0)$ be covered with a finite number of balls of radius $<r_0$

Consider an infinite dimensional Banach space $X$. Let $B(0,r_0)$ be the ball with radius $r_0$. We know that the ball $B(0,r_0)$ is not relatively compact, so it is not totally bounded. This implies ...
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3answers
27 views

Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
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21 views

Compact Operators: Adjoint

Given Banach spaces $E$ and $F$. Consider a bounded operator: $$T:E\to F:\quad\|T\|<\infty$$ As a result due to Schauder: $$T\text{ compact}\iff T'\text{ compact}$$ How to prove this fact?
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1answer
31 views

Banach space of p-Lipschitz functions

Given $p\in\mathbb{R}$, consider the space: $$ Lip(p) = \left\{f:[0,1] \longrightarrow \mathbb{R} : \mbox{ $f$ is $p$-Lipschitz} \right\}$$ i.e.: there is $M>0$ such that ...
2
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3answers
55 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
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3answers
57 views

What is abstraction of direction in considering vectors such as used in Engineering & Physics?

In the use of vectors of engineering and physics, we encounter objects that obey the axioms of a vector space but also have two new attributes of length (or, magnitude) and direction (e.g. direction ...
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31 views

is the complexification of a finitely strictly singular operator itself FSS?

Let $X$ and $Y$ be real Banach spaces, and let $X_\mathbb{C}$ and $Y_\mathbb{C}$ denote their respective complexifications. Suppose $T:X\to Y$ is a bounded linear operator which is finitely strictly ...
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27 views

Minkowski integral inequality in Banach space

Let $X$ be a Banach space of all measurable functions on $\mathbb{R}^d$ with the property: for any non-negative increasing sequence $\{f_n\}\subset X$, we have $\|\lim_{n\to \infty} f_n\|_X=\lim_{n\to ...
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1answer
31 views

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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Prove that $(\mathbb R^n,\|\cdot\|_3)^*= (\mathbb R^n,\|\cdot\|_{1.5})$ with full details

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 ...
3
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1answer
43 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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19 views

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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13 views

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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1answer
25 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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1answer
15 views

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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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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41 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
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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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35 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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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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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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51 views

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. ...
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148 views

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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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
64 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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28 views

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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32 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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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 ...
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52 views

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
46 views

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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1answer
32 views

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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1answer
49 views

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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1answer
28 views

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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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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26 views

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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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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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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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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46 views

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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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 ...
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130 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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46 views

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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1answer
45 views

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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57 views

About a technique used in the proof of Hahn-Banach Theorem

Recall Hahn-Banach (cf. Kreyszig's book) : If $X$ is a real vector space with a sublinear functional $p$ and if $f$ is linear on a subspace $Z$ with $p(z)\geq f(z),\ z\in Z$, then there exists an ...