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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Let $X$ be a Banach Space.Prove that, $X$ is strictly convex iff every points of $S(X)$ is exposed points of $B(X)$.

Some definitions- $X$ is said to be strictly convex or rotund if for all $x,y\in S(X),\ x\ne y$ we have $\left\lVert\frac{x+y}{2}\right\rVert<1$ A point $x_0\in B(X)$ is said to be a exposed point ...
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How to prove that the convergence of $(f_n)$ is uniform on compact sets?

I'm reading Theorem 0.9 in this lecture note. Below is my attempt where I got stuck at the end. Could you elaborate on how to finish the proof? Let $C$ be an open convex subset of a Banach space $X$....
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$X$ is strictly convex (rotund) iff for all $x,y\in S(X)$ with $x\ne y$ we have $\lVert x+y\rVert <2$

A Banach Space $X$ is said to be strictly convex or rotund if for all $x,y\in X$ we have $\lVert x+y\rVert<\lVert x\rVert+\lVert y\rVert$ unless $x,y$ are multiple of each other. We have to prove ...
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If $\mathcal{F}$ is pointwise bounded, then $\mathcal{F}$ is locally equi-Lipschitz and locally equi-bounded

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X$ be a Banach space and $\mathcal{...
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$ int(\overline{T(B(0,1))})\neq \emptyset $ in Banach space

Suppose that $X,Y$ are Banach space and $T:X\rightarrow Y$ is linear, continuous and overjective then $ int(\overline{T(B(0,1))})\neq \emptyset $ This is part of a bigger theorem, I have been trying ...
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Example of discontinuous convex l.s.c. function on an open convex subset of an incomplete normed space

I'm reading Proposition 0.7. in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is continuous on $C$. (b) If ...
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A Hausdorff's distance convergence

Let $E$ be a Banach space and $(C_n)_n$ be a sequence of nonempty closed sets of $E$. Let $C:=\bigcap_{n \in \mathbb{N}} C_{n}$. We assume (here) that $C$ is not empty. Let $d_H$ be the Hausdorff ...
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Finding metric projection mapping

Let $X$ = $c_0$, with supremum norm and $W$ = $\{ \{x_n\}_{n \geq 1} \in X$ $|$ $x_1=0\}$. Then how to find $P_W(x)$. Note that, $P_W(x)$ = $\{y_0 \in W : ||x-y_0|| = \inf ||x-y||,$ $ y \in W \}$. In ...
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Is the trace of this operator finite? [duplicate]

Let $H$ Hilbert space and let $Q\colon H \to H$ be a linear, self-adjoint, positive, trace-class operator. Let $X\colon H \to H$ be a linear, self-adjoint, positive operator. Does it follow then that $...
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Show that $\{x\in\mathbb{C}^{\mathbb{N}}: \lim_{n\to\infty} \frac{x_n}{a_n}=0\}$ is a Banach space

Consider $a:=\{a_n\}_{n=0}^{\infty}\in\mathbb{R}^{\mathbb{N}}_{+}$ a decreasing sequence with $a_n\to 0$. Consider the following space of sequences $$ X_a:=\left\{x:=\{x_n\}_{n=0}^{\infty}\in\mathbb{C}...
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Question regarding Cauchy Sequences on a Banach Space [closed]

I am currently in front of the following two problems and need help. Let $\mathcal{X} = (\mathcal{V}, \left\| . \right\|)$ be a Banach space. Let $\{x_n\} \subset \mathcal{X}$ be a Cauchy sequence in ...
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If $A$ is a generator of a strongly continuous semigroup and $B$ a bounded linear operator then $A+B$ generate a strongly continuous ssemigroup?

Let $X$ Banach and $A \colon D(A) \subset X \to X$ be a generator of a strongly continuous semigroup $e^{At}$. Let $B \in L(X)$ (linear bounded opearator on $X$) we then know that $B$ generates a ...
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Weak* separability of dual unit ball of D[0,1]

Let $D[0,1]$ be the space of all right-continuous left-limited functions $f\colon [0,1]\to \mathbb{R}$ equipped with the supremum norm $f\mapsto \|f\|_\infty = \sup_{t\in[0,1]} |f(t)|$. This is a non-...
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Equivalent Lebesgue integrability

I'm reading section "Integration of $\overline{\mathbb{R}}$-valued functions" on page 103 from Amann's texbook Analysis III. The decomposition of an $\overline{\mathbb{R}}$-valued function ...
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Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

In https://en.wikipedia.org/wiki/Direct_integral I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras. Suppose that I want to define a direct integral on either ...
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What will be the upper bound of covering number of $r B_{n}^2$

I am very curious to know The covering numbers of the r-radius Euclidean ball $B_{n}^2$ for any $r$ > 0: so $r B_{n}^2$ is a Euclidean ball with radius $r$ Have read this but unable to find. Can ...
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Prove that if $f\in C^{r}$ then the map $g(u,x):=(df)_u(x)$ is of class $C^{r-1}$

Let $X,Y$ be two finite dimensional $\mathbb{R}$-banach spaces and $U\subseteq X$ an open subset. Suppose that $f:U\to Y$ is of class $C^{r}$ with $r\geq 1$. Show that the map $g:U\times X\to Y$ given ...
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Boundedness of a subset of a Banach Space

Let $X$ be a Banach space. Show that: (1) A set $A^{\prime} \subset X^{\prime}$ is bounded in $X^{\prime}$ if and only if $M_{x}=\left\{f(x): f \in A^{\prime}\right\}$ is bounded for every $x \in X$. ...
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Is the orthogonal complement projector a projection operator?

Let $X$ Hilbert space and $U \subset X$ convex closed. Then we can define the projection operator on $U$ $P \colon X \to X$ such that $P x \in U$ is the orthogonal projection of $x \in X$. Moreover $P$...
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Books/online notes for analytic maps in Banach spaces

Can anyone suggest some good books/online notes on (real) analytic maps between Banach spaces? I am looking for the basic definitions and the implicit function theorem in this setting. Thanks!
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Understanding the proof of Fuglede's theorem

I am trying to understand this proof of Fuglede's theorem on wikipedia : Everything was making sense until the last sentence: "Considering the first-order terms in the expansion for small λ, we ...
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Banach space with continuous multiplication is a Banach algebra

Consider a Banach space $(\mathcal{A},\|\cdot\|_{\mathcal{A}}$) and assume that $\mathcal{A}$ is also an associative algebra with a unit element $I\in \mathcal{A}$ such that $\|I\|_{\mathcal{A}}=1$. ...
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A isomorphism between $\mathcal{L}(U,V)$ and $\mathcal{L}(U) \times \mathcal{L}(V)$.

Let $U$, $V$ be two Banach spaces and define the spaces $$ \mathcal{L}(U\times V) = \{T : U\times V \rightarrow U\times V : T \text{ is linear and bounded}\}, $$ $$ \mathcal{L}(U) = \{T : U \...
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Relationship between Banach Space and Measurable Space

Motivation I am asking this question because in the definition of a Martingale on Wikipedia they define a random variable $f:\Omega\to S$ over a Banach space S. However typically one defines a random ...
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Prove the inequality in Banach space

Let $X$ be a Banach space and let $(x_n)_{n=1}^\infty$ be a sequence in $X$ such that, for some $x \in X$, $$ \lim_{n \rightarrow \infty} \ell(x_n) = \ell(x) \text{ for all bounded linear functional }...
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How to prove that normed space is complete?

$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
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How to prove boundedness of a shift operator of an infinite sequence in two directions

Consider X = $l^{p}$ (Z; C) = { (. . . , $x_{−1}$, $x_{0}$, $x_{1}$, . . .) | $x$$_{k}$ ∈ C} I want to show that the right shift operator $Sr$ | ($Sr\hspace{1 mm}x)_{k}$ = $x_{k−1}$, $x$ = $(x_{k})_{...
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Lebesgue's criterion for Riemann-integrable functions with values in a Banach space

Does anybody know a definition of Riemann-integrability for functions with values in an arbitrary (!) Banach space for which Lebesgue's criterion holds which says that a bounded function is Riemann-...
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operator between Banach spaces and its boundeness

I am trying to understand the proof of the following: Let $T:X\to Y$ be an operator between two Banach spaces $X$ and $Y$ and $T$ is linear. Let $\mathrm{dim}\,X=n\,<\infty$. Then $T$ is bounded. ...
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ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$ \Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]). $$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)| $ for $f \in C([0,1])$...
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Rolle's Theorem for functions defined on Banach space

In J.Dieudonne's "Treatise on analysis, vol. 1" Chapter 8.2, there's a problem which asks to prove Rolle's theorem for a function defined on a Banach Space. The problem is as follows: Let f ...
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1 answer
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On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
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Norm convergence of series in banach space

Let $\mathfrak{X}$ be a Banach space and $\left(\rho_{n}\right)_{n \in \mathbb{N}}$ be a sequence in $\mathfrak{X}^\#$. Prove that the following assertions are equivalent: For each norm-convergent ...
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$0$ is an interior point of $\overline{T(S_1)}$, where $T(S_1)$ is the closure of $T(S_1)$

If $B$ and $B′$ be Banach spaces and $T$ be a continuous linear Transformation from $B$ onto $B′$. Let $S_1$ be an open sphere of radius $1$ in $B$. Then show that $0$ is an interior point of $\...
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Is $L^p(\mathbb{R})$ the $\ell^p$-direct sum of $L^p([0,1])$?

Given a sequence of Banach spaces $(A_n)_{n\geq 1}$ and $1\leq p\leq\infty$, we can form the direct sum $\bigoplus_{n\geq1}A_n$, where the norm of a sequence $(a_n)_{n\geq1}$ is defined to be the $\...
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Closed mid-point convex set and it's usefulness

$(X, \|•\|) $ be any normed space and $E\subset X$ . $(I)$ For every sequence $(x_n), (y_n) \in E$ with $x_n\to x $ and $y_n\to y $ in the space $(X,\|•\|) $ implies $\frac{x+y}{2}\in E\space \space $...
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6 votes
2 answers
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dense subspace of $L^2$ that is disjoint to $L^p$

I wonder if it is possible to have a dense subspace $U \subseteq L^2$ that is disjoint to $L^p$ for some $p\neq 2$. I would expect that such $U$ exists, but I'm stuck finding an example.
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The space of functions vanish at infinity

Why we care about these two Banach spaces $$C_0(\Omega):=\{ f \in C(\Omega) \colon \forall \epsilon>0 \text{ there exists a compact } K_\epsilon \subset \Omega \text{ such that }\\|f(s)|<\...
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1 vote
1 answer
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The space of finite signed measures with the total variation norm is a Banach space

I'm trying to grasp the idea of Hahn decomposition by proving below result. Could you verify if my attempt is fine? Let $X$ be a topological space and $\mathcal M:=\mathcal M(X)$ the space of all ...
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Identify a separable Hilbert space with the space $\mathcal l^2$

Let $X$ separable Hilbert space with orthonormal basis $(e_i)$. Then any $x \in X$ can be written as $$x=\sum_{\mathbb N}x_i e_i$$ for $x_i \in \mathbb R$ and thus it is quite natural to identify $X$ ...
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Example where spectrum of $AB \neq BA$ in a general unital Banach Algebra.

I am looking for an example where the spectrum of $AB \neq BA$ in a general unital Banach algebra. Now I have shown that the spectrums coincide if we ignore 0, so the only way the spectrums can be ...
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3 votes
1 answer
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A sort of converse of Banach-Steinhaus theorem.

$(X, \|•\|) $ and $(Y, \|•\|') $ be two normed space. $\begin{align} {\scr{B}}{(X, Y) }&=\{T\in {\scr{L}}{(X,Y)}: T \text{ is bounded } \}\end{align}$ $\|T\|_{op}=\sup\{\|Tx\|':\|x\|\le 1 \}$ ...
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1 answer
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The space of Hilbert-Schmidt operators form a Banach space

I am having trouble proving the following result: Show that the space $X$ of bounded operators on a separable Hilbert space into itself for which the Hilbert-Schmidt norm is finite, is a Banach space ...
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Proving the uniform convexity of $L^p$ for $1 < p \le 2$

We are asked to show the uniform convexity of $L^p$ for $1 < p \le 2$ using the following inequality: For all $1 < p < \infty$, there is a constant $C$ such that $|a - b|^p \leq C(|a|^p + |b|...
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Difference between strong and uniform operator topologies

I am currently a little confused regarding the difference between the strong and uniform operator topologies. I know that, if $T , T_n : V \rightarrow W$, then $T_n \rightarrow T$ in the uniform ...
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2 votes
1 answer
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What will be the next step to ensure that the pointwise convergence implies convergence in the operator norm?

$(X, \|•\|) $and $(Y, \|•\|') $ be two normed space and $\begin{align} {\scr{B}}{(X, Y) }&=\{T\in {\scr{L}}{(X,Y)} : T \text { is bounded } \}\end{align}$ $\|T\|_{op}=\sup_{\|x\|\le1}\|Tx\|' $ I ...
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1 vote
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Exercise 5.16 from Brezis' Functional Analysis

Suppose $H$ is a Hilbert space with scalar product $(\cdot , \cdot )$ and norm $| \cdot |$. Let $V \subseteq H$ be a dense linear subspace and say it has its own norm $\| \cdot \|$ such that $V$ is ...
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Example of the continuity of the Fredholm index

Consider the space of square-summable sequences $\ell^2 =\{(a_0,a_1,\ldots) : \sum_{n=0}^\infty |a_n|^2<\infty\}$. Let $I$ denote the identity on $\ell^2$ and define the unilateral shift operator \...
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1 answer
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Convex and Lipschitz implies "bounded" subgradient in Banach spaces

In this question: Lipschitz implies bounded gradient it is shown that if $f: \mathbb{R}^n \to \mathbb{R}^n$ is convex and L-Lipschitz the gradient is bounded by L. I wonder if this holds in Banach ...
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2 answers
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Norm of a map in Banach space

Let $X$ be a banach space and $f:(\Omega, \mathcal{F}, \mathbb{P}) \to (X,\mathcal{B})$ measurable function. I want to define $||f||_{L_1(X)}$. I saw this definition in a paper: $$||f||_{L_1(X)} = \...
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