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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Norm of a polynomial and its polar

I've been struggling so hard with this problem and I can't find a solution. The problem says: Let $(H,\lVert\cdot\rVert)$ be a complex Hilbert space, and let $H\oplus_\infty\mathbb{C}:=H\times \...
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Is the span closed?

Let X be a normed space. Let $v_1, v_2... v_n$ be vectors in X. When is $Span({v_1, v_2, ... v_n})$ closed? This question is motivated by a question which I had on a problem sheet (this wasn't the ...
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Combing Compact Operators and Pre-Compactness

Background I have had a lecture in my class on what it means for an operator to be compact i.e. for any bounded B, $\overline{T(B)}$ is compact. Now I have also had a theorem about classifying ...
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Is $\mathcal{B}(L^p,L^1)$ strictly bigger than $L^q$?

Let $(\Omega, \mathscr{F}, \mu)$ be a $\sigma$-finite measure space and $1\leq q \leq \infty$. Denote by $p$ the conjugate exponent of $q$. We know, given a $g\in L^q (\Omega)$, \begin{equation} ...
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Space of Hilbert-Schmidt operators

Aim. I need to show that the following is a Hilbert Space. Space X of bounded operators on a separable Hilbert space into itself for which the Hilbert- Schmidt norm is finite. Proof. I can first ...
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How to prove that $W^n_p[0,1]$ is a Banach space?

Problem. Let $1\le p<\infty$ and $n \ge 1$ and let $W^n_p[0,1]=$ the functions $f:[0,1]\to \Bbb{C}$ such that $f$ has $n-1$ continuous derivatives, $f^{(n-1)}$ is absolutely continuous, and $f^{(n)}...
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Weak vs. Weak* Convergence of Bounded Nets

Let $X,Y$ be Banach spaces. In V. Paulsen's book on completely positive maps, he shows that $B(X,Y^\ast)$ is a dual space as follows: For $x \in X, y \in Y$ let $x \otimes y \in B(X, Y^\ast)^\ast$ be ...
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Show that the vector space $l_1$ with the norm defined by $\| (a_1,a_2,..)\|= \sum_{k=1}^{\infty}|a_k|$ is a Banach space. [duplicate]

Show that the vector space $l_1$ with the norm defined by $\| (a_1,a_2,..)\|= \sum_{k=1}^{\infty}|a_k|\leq\infty$ is a Banach space. My work- Let $(a_{i})=(a_{{i}{1}},a_{{i}{2}},a_{{i}{3}}...)$ be a ...
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Compute norm in a Banach space

Let $I = [0, 1] \subset \mathbb{R}$ and the scalar field is $\mathbb{R}$. For a Banach space $C(I)$, let $\Lambda(f)=\int_{0}^{1}\left(9 t^{4}-18 t^{3}+11 t^{2}-2 t\right) f(t) d t$ I would like to ...
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Determine the norm of the continuous linear operator $T:L^1[a,b] \to L^1[a,b]$.

I have encountered an exercise. Let $K(x,y)$ be a measurable function on $[a,b]\times [a,b]$. The function $I:y\in [a,b] \mapsto \int_a^b |K(x,y)|\, \text{d}x \in [0,+\infty]$ belongs to $L^\...
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'Orientation-preserving' linear isometry

Let $X$ be a Banach space (on $\Bbb{R}$ or $\Bbb{C}$), $A:X \rightarrow X$ satisfy that $A\theta=\theta$ ($\theta$ is the zero element) and $||Ap-Aq||=||p-q|| \; \forall (p,q) \in X^2$. We call $A$ a ...
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Folland 5.49(b): Bounded subsets in $X$ (resp. $X^*$) are nowhere dense in the weak (resp. weak*) topologies

I'm reviewing for quals and I came across a problem on weak topologies I can't seem to put together (it could be because it's 11pm!): Let $X$ be an infinite dimensional Banach space. Then every (norm)...
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Metric Linear Space

I'm not sure if this is the right place to ask and would appreciate it if someone directed me elsewhere if this is the wrong place to ask. I'm looking for exercises where I have to prove that a space ...
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38 views

Application of Banach-Steinhaus theorem to judge on continuity.

Let $X$ be a Banach space and $T: X \rightarrow l^1$ linear map $T(x) = (T_n(x))_{n=1}^\infty$. Also for all sequences $a_n \in \{-1, 1\}$ funcional: $$F: X \ni x \rightarrow\sum_{n=1}^\infty a_n T_n(...
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Given $ST$ Fredholm then $T$ is Fredholm iff $S$ is , in Banach spaces

Let $X,Y,Z$ be Banach spaces. $T\in B(X,Y), S\in B(Y,Z)$ and $ST$ is Fredholm. Show that $T$ is Fredholm iff $S$ is Fredholm. I saw here possible answers for a similar question however it did not use ...
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Regularity of the fixed point of a Kernel

Take $S\subset \mathbb R^n$ bounded and be $k(\cdot|s)$ a probailiplity density kernel, i.e. $$\int_S k(s'|s)\ ds'=1\qquad s-a.e.$$ What are the hypothesis I have to put on $k$ so that, for any $b(s)\...
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Homogeneous Besov space $\dot B^{-\sigma}(\mathbb R^d)$ is a Banach space

Let $\sigma > 0$ and $\theta \in S(\mathbb R^d)$ (Schwartz function) be fixed, where $\theta$ is such that its Fourier transform $\hat{\theta}$ is compactly supported, $0 \leq \hat{\theta} \leq 1$ ...
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Clarification Regarding Banach space and Baire Category Theorem

New to Baire Category. From a remark: If we suppose that $Y$ is an infinite dimensional subspace of a Banach space $X$, and $Y$ has a countable (Hamel) basis, then one can show that $Y$ is first ...
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A characterisation of uniform convexivity

I want to prove that a Banach space is uniformly convex if and only if $$\|x_n\| \to 1, \quad \|y_n\| \to 1 \quad \text{and} \quad \left\|\frac{x_n+y_n}{2}\right\|\to1$$ implies $$ \|x_n - y_n \| \to ...
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Separability of unit ball in strong operator topology

Let $X$ be a separable Banach space. Let $B_1(X)$ be the set of all bounded linear operators $X \to X$ with operator norm $\leq 1$. Does $B_1(X)$ have to be separable in the strong operator topology?...
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How can I understand that the two characteristics of the $\omega$-limit set are equal?

I have a question about equations (2.1) and (2.2) in the following paper: https://homepages.warwick.ac.uk/~masdh/IDDS.pdf The two equations each give a definition of the omega-limit set, but I do not ...
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1answer
47 views

Unique element with minimal norm in closed convex subset

I have shown that for $X$ a hilbert space, $K$ a non empty closed convex subset of $X,$ that $K$ admits a unique element of minimal norm. I am now looking for an example of a Banach space where this ...
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Lifting of Reflexivity

I am looking for a proof (the simple the better) of the Theorem stating that $L_p(E)$ is reflexive if and only if $E$ is reflexive. Here is $E$ a Banach space and $L_p(E)$ is the Lebesgue-Bochner ...
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Fréchet derivatives for $f: \mathbb{R}^{d\times d} \rightarrow \mathbb{R}^{d\times d}_{sim}$

I am starting my studies at Fréchet derivatives and I saw the exercise below: Let $\mathbb{R}^{d \times d}$ with the usual operator norm. We know it is a Banach space. Let $\mathbb{R}^{d\times d}_{...
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Proving every linear operator in the dual space is compact.

How can I prove that for $X$ being a Banach Space, every $\phi \in X^*$ is compact?
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Supersets of unit ball in infite dimension

Let $X$ be a Banach space with infinite dimension. Is it true that every superset of the unit ball $B_X(0)$ in $X$ is not compact. In finite dimensions this is not true. I couldn't find any ...
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Why is the space of linear operators to a Banach Space complete

I know this has been answered before, but I wanted to understand a particular proof. We have a space X and a Banach Space Y. We take a sequence of bounded linear operators from X to Y i.e. $T_n$ which ...
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37 views

Weak* convergence and weak convergence in X*

Suppose $X$ is a Banach space and $\{f_n\}$ is a sequence in $X^*$ such that $f_n$ converges weak* to $f\in X^*$ (meaning $\lim_{n\to\infty}f_n(x) = f(x)$ for all $x\in X$), and $\lim_{n\to\infty}\...
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The space $\ell^p(a_n)$

Let $a_n$ be a positive sequence in $\mathbb{R}$ and $$ \ell^p(a_n)=\{ x=(x_j) : \sum_{n=1}^{\infty}{a_n} \vert x_n \vert^p < \infty\} $$ with the norm $\Vert x \Vert = \Big( \sum_{n=1}^{\infty}{...
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37 views

$Y$ is Banach; if $X$ is a topological space, $C_b(X,Y)$ is Banach [closed]

Let $X$ is a topological space and $Y$ is normed space. $C_b(X,Y)$ is Banach iff $Y$ is Banach.
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Prove the continuity of the map defined on locally compact Hausdorff space

A ternary Banach space $(A,[,])$ is complete normed space with a linear ternary map $[,]:A\times A\times A \to A$ satisfying $\vert \vert[a,b,c] \vert \vert \leq \vert \vert a \vert \vert \vert \vert ...
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“Doubly Lipschitz” function composition

Let $U$ and $V$ be Euclidean spaces. Let $X\subseteq U$ and $Y\subseteq V$ be open, bounded subsets. Let $G:=\operatorname{BLip}(X\times Y,U)$ be the (Banach) space of bounded Lipschitz functions from ...
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37 views

Suppose $f \in L^1$. Show that $\lim_{n \to \infty} \frac{1}{2n} \int_{-n}^{n} fdx = 0$

Suppose that $f \in L^1$. Show that $$\lim_{n \to \infty} \frac{1}{2n} \int_{-n}^{n} fdx = 0$$ I have an idea here. I believe I want to show that $\int_{-n}^{n} f dx$ is finite. If it is finite, then ...
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1answer
28 views

$L^p + L^r$ is complete

From Folland, "Real Analysis, Modern Techniques and Their Applications", Section 6.1 exercise 4. Let $1 \le p < r \le \infty$. Show that $L^p + L^r$ with norm $\|f\| = \inf\{\|g\|_p + \|...
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Unique extension of a bounded linear operator : Reference request

Does someone know a textbook which states the theorem considered in this question? Preferably, such a book should be released recently rather than many years ago as I'd prefer a source which does not ...
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Homogeneous analytic function on Banach space

Definition: Let $X$ be a complex Banach space. We say $f: X\rightarrow \mathbb{C}$ is holomorphic if it satisfies the following: 1.$f$ is locally bounded. That means given $x\in X$, there exists an ...
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50 views

Example of function that is Gâteaux-differentiable but not Fréchet-differentiable

I am looking for an example of a function that is Gateaux-differentiable but not Fréchet-differentiable. I know that there is a lot of example of function $f: \mathbb R^2 \to \mathbb R$ that satisfies ...
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Show that $\{f_n\}$ is a Cauchy sequence and that $\{f_n\}$ is not a converging sequence.

Can someone improve this answer or is there another way of solving it? Help will be much appreciated Q. Define $\|f\|_1=\int^1_{-1}|f(x)|dx$ for $f \in C([-1,1])$. Let $$f_n(x)= \begin{cases} -1&-...
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Ordering of fixed points

Suppose one has two real functions $y = f_a(y)$ and $y = f_b(y)$ for which the contraction mapping theorem holds, such that there exist unique fixed points $y^*_a$ and $y^*_b$, respectively. Are there ...
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1answer
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Question about the proof of existence of complemented subspaces of $L^1$

I have a question about the proof from the book ''Banach Spaces of Continuous Functions as Dual Spaces'', stated as Proposition 2.4.8, that is The Banach space $\ell^1$ is isometrically isomorphic to ...
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1answer
33 views

Complete normed space and convergences [duplicate]

Let $X$ be a normed space. Prove that $X$ is complete if and only if $\Sigma_{n=1}^{\infty} x_n$ exists for any sequence $\{x_n\}$ that satisfies $\Sigma_{n = 1}^\infty \|x_n\| < \infty$. Here $\...
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1answer
31 views

Showing that $F(x) = x + f(x)$ defines a homeomorphism when $f : E \to E$, and where $E$ is a Banach space.

Let $E$ be a Banach space and $f : E \to E$ a contraction. Show that the equation $F(x)=x+f(x)$ defines a homeomorphism $F:E \to E$ that is Bilipschitz. Since $f$ is a contraction the following to ...
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103 views

$\|x_0 − y_0\| = \operatorname{dist}(A, B)$ [closed]

Let $X$ be a reflexive Banach space and let $A, B$ be non-empty, closed, and convex subsets of $X$. If $B$ is bounded, prove that there exist $x_0$ in $A$ and $y_0$ in $B$ with $\|x_0 − y_0\| = \...
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29 views

Fréchet derivative of a (nonlinear) differential operator

The question may not be too well-posed, but loosely speaking, suppose $L:W^{1,p}(\mathbb R)\to L^p(\mathbb R)$ is a (possibly nonlinear) first order differential differential operator, such that all ...
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225 views

Understanding Lang's Proof of Fubini's Theorem

This question concerns the proof of Theorem 8.4 (Fubini's Theorem part 1) on page 162 in Lang's real and functional analysis book. To understand the proof I need to give following background from the ...
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58 views

How to prove $L^1[a,b]$ is weakly sequentially complete in a more elementary way?

I have referred to GTM233, but the proof on it is too difficult for me. And I know that $\ell^1$ is weakly sequentially complete since it has the Schur property. I wonder does it hold for $L^1[a,b]$? ...
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139 views

About the spectrum $\sigma(T)$ [duplicate]

Let $X$ be a Banach space and $T\in B(X)$. Show that $$\bigcap_{K\in K(X)} \sigma(T+K)= \sigma(T)\setminus \{\lambda : \lambda I-T \text{ is a Fredhold operator with index zero}\}.$$ I looked at the ...
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2answers
47 views

Norm of Linear operator as a limit

It is known that a linear operator on Banach space not necessarily attains its norm (meaning that there is no element $x$ s.t. $\|Tx\| = \|T\|$). I would like to clarify that there is a sequence that ...
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1answer
46 views

Why is $C^{k}([a,b],||\cdot||)_{C^{k}}$ a Banach space? [closed]

If $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ and $||\cdot||_{C^{k}}=\sum_{i=0}^{k}||f^{(k)}||_{\infty}$, why is this a complete norm on $C^{k}([a,b]\longrightarrow\mathbb{K})$?
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31 views

Unbounded linear operator with closed null space

Let $T$ be a linear operator between Banach spaces $\mathscr{X}$ and $\mathscr{Y}$ which is defined everywhere in $\mathscr{X}$. Could $T$ have a closed null space $N(T)=\{x \in \mathscr{X}|Tx=0\}$ ...

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