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Questions tagged [banach-spaces]

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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Does "shift of terms" in a Faber–Schauder series expansion of f ∈ C[0,1] produce an element of C[0,1]?

Question Let $\{e_n : n=0, 1, \dots \}$ be a Faber–Schauder basis of $C[0,1]$ (see: Example 4.1.11 in [1]). Is the following function well-defined: $$ \begin{align} T \colon C[0,1] \times \mathbb{R} &...
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Homotopy between bounded linear operators on Banach spaces

In my course of Functional Analysis, talking about continuity method and index of Frendholm operators we construct homotopy between operators. However i encountered some difficulties to connecting ...
Manuel Bonanno's user avatar
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Conditions for the existence of eigenvectors for a positive operator on a Banach Space

Suppose $X$ is a Banach space and $K \subset X$ is a normal cone with interior. Suppose further that $A$ is a non-compact strongly positive operator, $A(K) \subset$ int$K$. It is well known that under ...
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Let $X$ be a Banach space and $M \subset X^*$. Prove: $M$ is bounded if and only if for every $x \in X$ {$\varphi(x) \mid \varphi \in M$} is bounded.

Let $X$ be a Banach space and $M \subset X^*$. Prove that the set $M$ is bounded if and only if for every $x \in X$, the set {$\varphi(x) \mid \varphi \in M$} is bounded. Attempt: If $M$ is bounded, ...
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Estimation of the right metric tensor for a vector-space valued and Normally Distributed random vector

For a fixed value of $k \in \mathbb{N}$, let $P^{k}[\mathbb{R}]$ be the real vector space of all real polynomials of degree $< k$. Fix a real closed interval $I = [a, b]$ and $k$ distinct values $a ...
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Convex absorbing set

Let $X$ be a normed vector space. I'm aware a convex absorbing set $A\subset X$ might have empty interior. However, I wonder whether the following is true? Let $X$ be a normed space and $A\subset X$ ...
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Numerical radius norm is equal to the operator norm

Currently, I am studying numerical ranges and numerical radius of linear operators. As one of the references, I am studying the book ``Numerical Range: The Field of Values of Linear Operators and ...
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Weighted Bessel potential space is a Banach space?

I'm studying weighted $L^p$ spaces. In O. Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (Birkhäuser, Boston, Mass, 2010), weighted $L^p$ space is ...
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Bounded operators on the span of a set of vectors

Let $E$ be a Banach space and $S$ a subset of vectors in $E$. Suppose that a linear operator $T$ satisfies the property $$\|Tx\| \leq C \|x\| \text{ for all }x \in S.$$ Does this imply that $T$ is ...
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Problems understanding integration in a banach fixed point exercise [closed]

I have been studying for a math test and I have encountered the following problem: banach's fixed point theorem applied to finding a unique function I do not get the last step. I do not know how ...
valentina manfredi's user avatar
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Two sigma-algebras on $l^\infty$

Let $\mathcal{B}_w(l^\infty)$ denote the Borel $\sigma$-algebra generated by the weak topology on $l^\infty$. Let $Cyl(l^\infty)$ denote the cylindrical $\sigma$-algebra, that is, it is the $\sigma$-...
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Proof verification on Functional analysis excercise [closed]

Let $E$ be a Banach space and $T \in \mathcal{L}(E)$ with $\lVert T \rVert < 1$ . We denote by $T^0 = I$ where $I$ is the identity map of $E$ and $T^k = T \circ \overset{k}{\dots} \circ T$ for $k \...
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If X is a basis for Hilbert space H then exists Y, biorthogonal, with Y a basis for H. [closed]

I would like to prove that: given $X = \{x_n\}$, basis of the Hilbert space $H$, then it does exist $Y = \{y_n\}$ with is biorthogonal to $X$. Also, $Y$ is a basis for $H$. I found the first half in ...
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If the image of the canonical injection is closed, the space is complete

Let $E$ a normed space and $j: E \rightarrow E'', j(x)=\hat x$ where $\hat x : E' \rightarrow \Bbb{K}, \hat x(f)=f(x)$ . Then, $E$ is complete iff $j(E)$ is closed on $E''$ . I am trying to prove ...
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Reference for w*-to-norm continuity of the adjoint operator.

I am looking for a reference for the proof of the following theorem: $\textbf{Theorem}.$ Let $X$ and $Y$ be Banach spaces and $T:X\to Y$ be a bounded linear operator. Then $T$ is compact if and only ...
Jose M Barrientos's user avatar
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For $C^*$-algebras A and B is the map $A\times B \to A\otimes_{\min} B$ continuous?

For $C^*$-algebras A and B is the map $A\times B \to A\otimes_{\min} B$ continuous? Where $A\times B$ has product topology and $A\otimes_{\min} B$ the norm topology. If yes, a proof (or reference) ...
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spectrum of translation operator

Let $\displaystyle A: c \rightarrow c, (Ax)(n) = \frac{x(n+2)}{n^2}$. The task is to find point spectrum, continuous spectrum, and residual spectrum of $A$ and of adjoint operator $A^*$. $c$ is Banach ...
GeoArt's user avatar
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Closed subspaces of $L^1(\mathbb{R})$ that is not isomorphic to a subspace of a space with an unconditional basis

It is known that $L^1([0,1])$ does not admit any unconditional basis. Even more, $L^1([0,1])$ is not isomorphic to a subspace of a space with an unconditional basis. My question is the following ``...
Roddick Yu's user avatar
1 vote
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Relation between two norms

Let $p\ge 2$ and $w:\mathbb{R}^d\to \mathbb{R}_+$ be a weight function normalized such that $\|w\|_{L^1}=1$ (the examples I have in mind would be a Gaussian or a two-sided exponential for example). ...
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Positivity of the bidual operator

How is the fixed space of a bounded linear operator related to the fixed space of the bidual operator? Motivation of question: I have a strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Banach ...
Guest's user avatar
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How to prove that $A(f)(x) = \int_0^x e^t f(t) dt$ is bounded operator?

Let $A: C[0,1] \to C[0,1]$ be operator defined as $$ A(f)(x) = \int_0^x e^t f(t) dt $$ for all $x \in [0,1]$ and $f \in C[0,1]$ How to prove that $A$ is bounded? Also, similarly, how to prove that $$ ...
smth's user avatar
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Differential of non-Euclidean distance function

For a fixed value of $k \in \mathbb{N}$, let $P^{k}[\mathbb{R}]$ be the real vector space of all real polynomials of degree $< k$. Fix $k$ distinct values $x_1 < \ldots < x_k \in \mathbb{R}$ ...
Alberto's user avatar
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1 vote
1 answer
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Construction of an uncomplemented subspace of $\ell^{1}$

In the following I want to understand the construction of an uncomplemented subspace $\ell^1$ following the notes of "Topics in Banach Space Theory" $\textbf{Corollary 2.3.3.}$ The space $\...
Caratheodory's user avatar
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31 views

Generalization of Holder inequality for arbitrary finite products

Let $f_i \in L^{p_i}(X)$, i.e. $\|f_i\|_{p_i} < \infty$ for each $p_i$ where $\{p_i\}$ is a finite collection of reals in $[1, \infty]$ such that $\sum_{i=1}^N \frac{1}{p_i} = 1$. Then we have $$\|\...
Grigor Hakobyan's user avatar
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$\ell^{p}(I, Z \oplus Y) \approx \ell^{p}(I, Z) \oplus \ell^{p}(I,Y)$

In "The decomposition technique" by Pełczynski one uses that $\ell^{p}(I, Z \oplus Y) \approx \ell^{p}(I, Z) \oplus \ell^{p}(I,Y)$ where $\ell^p(I,X)$ is the infinite direct sum formed by ...
Caratheodory's user avatar
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95 views

Cauchy but not convergent sequence on $C[-1,1]$

Consider $C[-1,1] = \{ f: [-1,1] \rightarrow \Bbb{R}: f$ is continuous $\}$, the inner product $\langle f,g \rangle = \int_{-1}^1 f(x)g(x) \, dx$ and $\lVert \cdot \rVert$ the induced norm. Then, the ...
Daniel García's user avatar
3 votes
2 answers
80 views

If $E^*$ is separable and $\{x_k\}\subset E$ is bounded, then it has weak Cauchy subsequence.

Suppose that $E$ is a Banach such that $E^*$ is separable. Show that if $\{x_k\}_{k\in\mathbb{N}}\subseteq E$ is bounded in $E$ then it has a weak Cauchy subsequence. We define a Cauchy weak sequence ...
Nicolas Rodriguez's user avatar
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Is there an operator $T : X \rightarrow Y$ with closed range whose kernel is not complemented?

Let $T : X \rightarrow Y$ be a bounded linear operator between spaces $X$ and $Y$. Suppose that the range $T(X) \subseteq Y$ is closed. We know that bounded linear operators have closed kernel, and so ...
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1 answer
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Is $C(X)$ complete if $X$ is compact? [duplicate]

Let $X$ be a compact space, $C(X)$ be the metric space of continuous functions from $X$ to $\mathbb{R}$ with sup norm. Is $C(X)$ complete? I can prove it when $X$ is a metric space.
user46190's user avatar
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1 answer
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Equivalence of Definitions of Complemented Subspaces

I am seeking clarification on the equivalence of two definitions of a complemented subspace in functional analysis. $\textbf{Definition 1:}$ A closed subspace $W$ of a Banach space $V$ is said to be ...
Caratheodory's user avatar
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Are the famous operators S_n of Fourier series theory continuous?

In a previous Q&A of mine (this here) I mentioned what Katznelson calls homogeneous Banach spaces on the torus $\mathbb T = \mathbb R / 2\pi \mathbb Z $ (I mention him here because others define ...
Ulysse Keller's user avatar
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1 answer
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Dual of the direct sum of Banach spaces

Let us consider two Banach spaces $\mathbb E_1$ and $\mathbb E_2$. Let us equip the direct sum $\mathbb E_1\oplus\mathbb E_2$ with the two norm, i.e., $$ \left\Vert (\xi,\zeta) \right\Vert_2:=(\...
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Show which operators are unitary equivalent

I am trying to figure out which of these operators are unitary equivalent. $$1.\; t^3 x(t), \,\mathbf{L}^2[0, 1]\\ 2.\; (1-t)x(t),\,\mathbf{L}^2[0, 1]\\ 3. |t|x(t),\, \mathbf{L}^2[-1, 1]$$ Obviously ...
ed199957's user avatar
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0 answers
43 views

Intuition of compactness in function spaces

I understand compactness in $\mathbb{R}^n$. However, I do not know much about what sets of function spaces are compact. For example, what are the compact sets of the spaces $B([0,1], \mathbb{R})$ of ...
Power's user avatar
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1 answer
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Let $X$ be a Banach space, $A \in B(X)$ be injective, and $B: X \rightarrow X$ be a linear operator such that for every $f \in X^*$, $fAB \in X^*$.

Let $X$ be a Banach space, $A \in B(X)$ be injective, and $B: X \rightarrow X$ be a linear operator such that for every $f \in X^*$, $fAB \in X^*$. Prove that $B \in B(X)$. Attempt: To prove that $B \...
good12's user avatar
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A variation of Riesz Lemma

Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $||x||=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
Emerick's user avatar
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Can one construct an isometric embedding $\phi_p:\ell^p\to\ell^\infty$ without Hahn-Banach?

It is well-known (for example, see this question) that, as a consequence of the Hahn-Banach theorem, every separable Banach space can be isometrically embedded in $\ell^\infty$. In particular, for $1\...
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Compute the norm of a bilinear operator on the space of Lipschitz functions vanishing at $0$

Let $M=[-1,1]\subset ℝ$. A function $f: M → ℝ$ is a Lipschitz function if there exists a finite non-negative constant $C$ such that $|f(s)-f(t)| ⩽ C\|s-t\|$ for all $s, t ∈ M$. Denote the set of ...
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1 answer
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Has every homogeneous Banach space on the torus an operator S_n (Katznelson on Harmonic Analysis)?

In his book about Harmonic Analysis Katznelson defines homogeneous Banach spaces on the 1-dim. torus: in short, such a space is a linear subspace $B$ of $L^1(\mathbb T)$ with a norm denoted $||f||_B$ (...
Ulysse Keller's user avatar
2 votes
0 answers
58 views

Weirdest properties of Banach spaces?

This is soft question, but I've recently have become interested in the odd properties of Banach spaces after studying Banach spaces which do not admit a pre-dual. I also almost work exclusively with ...
Isochron's user avatar
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1 vote
2 answers
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Let (X, ||.||) be a Banach space and T: X $\rightarrow$ $X^*$ a linear operator with the property (Tx)(y) = (Ty)(x) for every x, y $\in$ X.

Let (X, ||.||) be a Banach space and T: X $\rightarrow$ $X^*$ a linear operator with the property (Tx)(y) = (Ty)(x) for every x, y $\in$ X. Let $(X, ||\cdot||)$ be a Banach space and $T: X \rightarrow ...
good12's user avatar
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1 answer
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Suppose $A$ is a bijective linear mapping and $Y$ is a Banach space.

Let X be a vector space and Y a normed subspace with norm $||_Y$. Let $A : X \rightarrow Y$ be a linear mapping. (a) Prove that with the prescription $||x|| = ||Ax||_Y$ for every $x \in X$, a norm is ...
good12's user avatar
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1 answer
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$T:\ell^{\infty}\rightarrow L^{p}$ is continuous (norm-to-norm), is it weak*-to-weak continuous?

Given norm-to-norm continuous map $T:\ell^{\infty}\rightarrow L^{p}$, $p\geq 1$. Does norm-to-norm continuous imply weak*-to-weak continuous? I have learnt that: if $T$ is norm-norm continuous then it ...
Chen Deng-Ta's user avatar
0 votes
1 answer
36 views

Monomials are NOT a (NOT necessarily Schauder) basis?

A basis for an infinite dimensional Banach space $X$ is a set of elements $\{e_k\}=B\subseteq X$, such that Each finite subset of $B$ is linearly independent. The closure of the span, i.e. $\{\lim_{N\...
Confuse-ray30's user avatar
2 votes
1 answer
77 views

Completeness of Y (normed tvs) in the proof of th. 2 in ch. V section 3 of Yosida's Functional Analysis ed 6

I'm stuck on the proof of theorem 2 in chapter V, section 3 of Yosida's Functional Analysis edition 6 (pages 140,141). Theorem 2 says : A locally convex linear topological space X is reflexive iff it ...
PTony's user avatar
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2 votes
1 answer
90 views

Coordinate-wise projections separate points in the closed span of a sequence

Let $X$ be a Banach space over $\mathbb K$ where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$, $(x_n)$ a linearly independent sequence (can be normalized with necessary or convenient) in $X$, $E$ ...
Emerick's user avatar
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3 votes
0 answers
32 views

Convex hull of cartesian product in general vector spaces

Is it true that $\text{conv}(X)\times\text{conv}(Y)\subset \text{conv}(X\times Y)$, where $X,Y$ are subsets of a (not-necessarily-finite-dimensional) vector space? If the answer is “no”, what if we ...
student566's user avatar
2 votes
0 answers
38 views

Can angles be defined by norms that are not induced by inner products?

Can angles be (well-)defined in a normed vector space where the parallelogram law does not hold? In other words, if the norm is not induced by an inner product in a normed space (say $L^1$ space), can ...
chaohuang's user avatar
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1 answer
51 views

Prove the compactness theorem for Radon measures by using Banach-Alaoglu theorem

I was reading the proof of the compactness theorem for Radon measures (Theorem 1.5.15) from Leon Simon's book: Geometric Measure Theory I was confused by the highlighted part. I hadn't learned the ...
OneLamp's user avatar
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0 answers
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Split and differentiable locally trivial submersion, and more confusions on a proof in John H Hubbard's book

The "split" means Let E, F be Banach spaces, A surjective linear map $f:E\rightarrow F$ is said to split if $ker f$ admits a closed complement. More generally, a submersion is a split ...
Kenny S's user avatar
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