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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A dual quasi-norm (or a generalized Cauchy Schwartz inequality for quasi-norms)

Let us assume that we have a quasi-norm $\Omega(\cdot)$ on $\mathbb{R}^{p}$, now is there something like a dual quasi-norm $\Omega^{*}(\cdot)$ such that a generalized Cauchy Schwartz inequality holds: ...
5
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1answer
44 views

$E$, $F$ Banach spaces, $A: D(A) \subset E \to F$ closed densely defined unbounded operator, does $N(A) = R(A^*)$?

Let $E$ and $F$ be two Banach spaces and let $A: D(A) \subset E \to F$ be a closed densely defined unbounded operator. Does it follow that $N(A) = R(A^*)^\perp$? Notation. Let $E$ and $F$ be two ...
2
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40 views

can a Banach space admit a subsymmetric basis AND a symmetric basis?

Definitions. A (Schauder) basis for a Banach space is called symmetric if it is unconditional and uniformly equivalent to all its permutations. It is called subsymmetric if it is unconditional and ...
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34 views

Hermitian Projections on $C[0,1]$

A projection on a complex Banach space $X$ is said to hermitian if its numerical range is real. Does anyone know an example of an hermitian projection on $C[0,1]_{\mathbb C}=C[0,1]\oplus i C[0,1]$?
0
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1answer
36 views

Range of any projection is closed.

Let $X$ be a Banach space and $P$ a projection. Show that the range of any projection is a closed subspace. Can I use the fact that a Banach space is complete and thus closed and that $P = P^2$ to ...
2
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24 views

Path of completely bounded maps has uniformly bounded cb norm?

If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...
6
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1answer
67 views

Question surrounding Exercise 3.12 of Brezis, function is convex and l.s.c. for the weak* topology.

Consider Exercise 3.12 of Brezis's Functional Analysis, Sobolev Spaces and Partial Differential Equations. Let $E$ be a Banach space and let $x_0 \in E$. Let $\varphi: E \to (-\infty, +\infty]$ be ...
1
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1answer
20 views

discrete convolution $f*g$ belongs to $\ell_\infty$, i.e. the sup norm is finite

Definitions. Fix any $\phi\in(0,1)$ and $\theta\in(0,1)$, and let us define functions \begin{equation}f(n)=\left\{\begin{array}{ll}n^{-\phi},&\text{ if }n\geq 1\\0,&\text{ ...
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22 views

Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows: Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...
0
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14 views

Meaning of amalgamated metric sum of $A_n$’s over $0$ and $d_n$ inherited from $\mathbb{R}^2$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
1
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1answer
52 views

How to proof that a finite-dimensional linear subspace is a closed set

Given a linear space V, a field F, a norm $||.||$ on V and a Base B. How do i proof that the sub-space span{$b_1,b_2,...,b_n$} where $b_i \in B$ is a closed set under the topology that is created ...
4
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105 views

If $f:X \rightarrow Y$ is a linear isomorphism between $X$ and $f(X)$, then show that there exists a continuous linear map from $Y^*$ onto $X^*$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
0
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1answer
79 views

A proof on the domain of semigroup

Let $T$ be an operator in a Banach space $X$ with the domain $D(T)$ equipped with the graph norm \begin{equation*} \|v\|_T=\|v\|_X+\|Tv\|_X \end{equation*} Assuming $\|v\|_T$ is a norm on $D(T)$, my ...
0
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1answer
23 views

Convex hull of $\{ \Vert x \Vert = 1 \}$ is closed in strictly convex space

I'm trying to show that the convex hull of $\{ \Vert x \Vert = 1\}$ is closed in a strictly convex Banach-space. I don't know how to tackle the problem. Are there any nice characterizations for a ...
3
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66 views

Corollary of the Hahn-Banach theorem.

Recall the Hahn-Banach Theorem in Normed Vector Spaces: "Let $X$ be a NVS, and let $W$ be a linear subspace of $X$. Then, for any $f_W\in W'$ there exists an extension $f_X\in X'$ such that ...
0
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25 views

$C^r(K,F)$ as a Banach space for $K$ compact, $F$ Banach space

Let $E$ and $F$ be Banach spaces and $K\subset E$ be compact. I want to understand what the "common definition" (if there is one) of the banach space $C^r(K,F)$ of $r$ times continuously ...
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1answer
31 views

If there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach

I have been asked to show that if there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach. I have shown it for $T$ a linear operator. But I can't use the ...
7
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1answer
103 views

Vector space that can be made into a Banach space but not a Hilbert space

Are there any (real or complex) vector spaces which can be made into a Banach space given a suitable norm, but cannot be given a norm that makes it a Hilbert space? I know that the parallelogram law ...
8
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1answer
85 views

Two Banach spaces, if and only if criterion for range of closed unbounded operator to be closed?

Let $E$ and $F$ be two Banach spaces. Let $A: D(A) \subset E \to F$ be a closed unbounded operator. How do I see that $R(A)$ is closed if and only if there exists a constant $C$ such ...
2
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1answer
29 views

Quotient map $\pi : X \rightarrow X / \mathrm{ker}(A)$ is open for a bounded linear operator $A$

I'd like to show: if $A : X \rightarrow Y$ is a bounded linear operator between Banach-spaces, then $\pi : X \rightarrow X / \mathrm{ker}(A)$ is a open map. I found a proof, which I do not really ...
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1answer
32 views

A finite dimensional normed vector space is a Banach Space.

I want to show that a finite dimensional normed vector space X over $\mathbb{K}$ ( $\mathbb{K}= \mathbb{R}$ or $\mathbb{C}$ ) is always complete by using the fact that X is isomorphic to ...
2
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16 views

sum of uniformly bounded projections acting on a Lorentz sequence space

It is known that for every $k\in\mathbb{N}$ there is $N_k\in\mathbb{N}$ such that every $N_k$-dimensional subspace of $\ell_p$, $1<p<\infty$, contains uniformly complemented copies of $\ell_2^k$ ...
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1answer
22 views

Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$

$a=a(x), b=b(x)$ are elements of $L^p(\Omega)$, $\Omega$ is bounded open subset of $R^n$. Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$ ?
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1answer
48 views

Relation between $L^1(\mathbb R)$ and $L^2(\mathbb R)$

I know that if $1\le p<q<\infty$ then $L^p\supset L^q $ and $l^p \supset l^q$. But what is the relation between $L^1(\mathbb R)$ and $L^2(\mathbb R)$? I guess there is no relation, i.e. ...
1
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1answer
19 views

Closed Image + dim ker $D<\infty$

I have some trouble to understand the proof of following Lemma: Lemma: $X$,$Y$ Banach spaces and let $D:X\to Y$ be a bounded linear operator. Following is equivalent: (i) dim kerD$<\infty$ and imD ...
1
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1answer
28 views

Applying Neumann series

Let $E$ be a Banach space. Let $A \in L(E)$, the space of linear operators from $E$. Show that the linear operator $\varphi: L(E) \to L(E)$ with $\varphi (T) = T + AT$ is an isomorphism if $\|A\| ...
2
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42 views

Density and Pointwise convergence imply strong convergence for Bi-orthogonal system?

Let $H=\ell^2$ and we denote $x = (x(k))_k$ for $x\in H$. Let $(e_n,f_n)_{n=1}^\infty$ be a bi-orthogonal system in $H$ , that is, $\langle e_m, f_n\rangle=\delta_{mn}$. For $x\in H$ we define, $$x_n ...
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34 views

Does an infinite dimensional Banach space always admit an infinite dimensional, separable subspace?

If that can't be achieved, what if the Banach space is reflexive?
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21 views

Space of linear combinations of bounded functions and $l_1$ Banach space

Let $I$ be an at most countable and non-empty set and $(h_i)_{i\in I}$ be a family of bounded functions $h_i:X\rightarrow \mathbb{R}$ with $\lVert h_i \rVert_{\infty}\leq 1$. In addition, $(w_i)_{i\in ...
9
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1answer
143 views

Characterization for the convergence of a series

Problem. Let $X$ be a topological spaces which is compact and Hausdorff, $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$, and suppose there exists a sequence $\{x_n\}_{n\in\mathbb N}\subset X$, such that ...
3
votes
2answers
40 views

Weakly convergence but not strongly - properties of limsup and liminf

Let $X$ be a Banach space and suppose we have a sequence $\{x_n\}$ which is convergent weakly but not strongly. Define $y_n:=\sum\limits_{k=1}^{n}x_k$. What we can say about ...
6
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1answer
69 views

Reflexive Banach space, strong closure and compactness.

Let $E$ and $F$ be two Banach spaces, and let $T \in \mathcal{L}(E, F)$. I have three questions. Assume that $E$ is reflexive. Is $T(B_E)$ strongly closed? Assume that $E$ is reflexive and that $T ...
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42 views

Direct sum of two closed subspaces of Banach space

We have $l^2$ as Banach space and $A$ and $B$ are closed subspaces of $l^2$ : $A=\{a\in l^2$| $a^{2n}=0$ }$ $ B={$a\in l^2| a^{2n}=a^{2n-1}/2n \} $. I have to prove $A+B$ is not closed, then i can use ...
0
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31 views

Is an unbounded function bounded on a bounded non-compact interval?

I'm a little confused about functions in the set of bounded continuous functions. For example, if we take the interval (0,1] and the function $f(x) = $\begin{cases} 0 & x \in ...
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10 views

What is the n-concavification of a Banach space?

I'm reading this paper about polynomials in Banach spaces and the authors use the notion of the n-concavification of a Banach space $X$ It is the first time that I encounter this concept. What is it? ...
2
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0answers
35 views

Derivatives in Banach spaces: question about Theorem 1.1.6 from Hormander

Let $U, V$ be Banach spaces, and suppose $X \subseteq U$ is open. let $L(U,V)$ denote the Banach space of continuous linear operators from $U$ to $V$. Suppose that $f$ and $g$ are continuous functions ...
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1answer
52 views

States on a $C^*$-algebra

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact. My problem is: Does the same hold if $A$ is not unital?
2
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1answer
41 views

What is the dual of the disc algebra viewed as a Banach space?

Let $A$ be the disc algebra, i.e., $A=\{f\in C(\bar{U}):f \text{ is holomorphic in }U\}$, where $U$ is the unit disc in the complex plane. The norm considered is the supremum norm. Are there any ...
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2answers
40 views

Shrink a banach space to a closed linear span of an into isometry

Theorem $2.2$ states that Let $X$ and $Y$ be separable Banach spaces and suppose that $f:X \rightarrow Y$ is an into isometry, then $X$ is linearly isometric to a subspace of $Y$. In the proof, ...
8
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1answer
107 views

Counterexample of polynomials in infinite dimensional Banach spaces

I'm trying to prove exercise I.3.B in Mujica's "Complex analysis in Banach spaces". DEFINITIONS: A map $P$ is an m-homogeneous polynomial from $E$ to $F$ if there is a m-linear map $A$ from ...
2
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28 views

Prove that every polynomial from a finite Banach space to another Banach space has an unique representation in terms of coordinate functionals

I'm trying to solve Problem 1.2.J in Mujica's "Complex Analysis in Banach Spaces". The problem states as follows: Let $E$ and $F$ be Banach spaces over $\mathbb{K}$, with $E$ finite dimensional. ...
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1answer
21 views

Double annihilator of subspace of $X'$ is its weak*-closure

Let $X$ be a Banach space with dual space $X'$. Let $N$ be a subspace of $X'$. Can anyone show me why the double annihilator of $N$ is its weak*-closure? By double annihilator I mean: annihilator ...
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1answer
36 views

Isometry between $L^{\bot}$ and $H/L$ where $H$ is hilbert and $L$ is a closed subsapce

Show $L^{\bot}$ and $H/L$(with $||\cdot||_{H/L}$) where $H$ is hilbert and $L$ is a closed subsapce are isometric, given $||\cdot||_H$ is defined by the inner product. By now I have considered ...
4
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1answer
42 views

The finite product of $L^p$ spaces is reflexive ($1<p<\infty$)

I am trying to understand the proof that the Sobolev Space $W^{1,p}$ is reflexive given in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. There it is used ...
2
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2answers
35 views

The Sobolev Space $W^{1,p}(I)$ is complete

I am trying to verify that Sobolev Space $W^{1,p}(I)$ is complete. This is the definition of $W^{1,p}(I)$: $W^{1,p}(I)= \{ u\in L^{p}(I) | \exists g \in L^p(I) :\int_{I}^{}u\varphi'= - ...
0
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0answers
30 views

Banach Space Direct Sum

Let $\mathcal B$ be Banach. Let $z \in \mathcal B$ be non-zero. Consider $A = \operatorname{span}(\{z\})$. Is it possible to find a closed subspace $C$ such that $$\mathcal B = A \oplus C?$$ If so, ...
0
votes
0answers
22 views

Convex combination of finitely many elements of the weakly converging sequence

Suppose $X$ is a metrizable locally convex Banach space and suppose the sequence $\{x_n\}$ weakly converges to some $x \in X$. I am trying to prove that there exists a sequence $\{y_m\}$ such that ...
1
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1answer
24 views

Isometry of the adjoint operator

Suppose that $T$ acts between Banach spaces $T: X \to Y$ and $T$ is surjective. One can define the adjoint operator $T^*:Y^* \to X^*$ by the formula $T^*(\varphi):=\varphi \circ T$. Is it true that if ...
-1
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2answers
97 views

Dual space of $L^\infty$ is $L^1$ with the weak-* topology?

A friend of mine found a book in which the author said that the dual space of $L^\infty$ is $L^1$, of course not with the norm topology but with the weak-* topology. Does anyone know where I can find ...
1
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1answer
32 views

Banach Space Closed Subspace

Let $ \mathcal B$ be a Banach Space. Fix $z \in \mathcal B$ with $z \neq 0$. Consider the set $$A :=\{y-z : y \notin \operatorname{span} \{z\}, y \in \mathcal B\}.$$ Is it true that $\alpha z \notin ...