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 isomorphism of duals implies isomorphism? [duplicate]

Does isomorphism (or isometric) of $X^*$ and $Y^*$ for $X,\:Y$ normed spaces (or banach) implies isomorphism (or resp. isometric) of $X$ and $Y$? I know that the other way around is true, but I never ...
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1answer
32 views

Absolute continuity in Banach spaces

Does absolute continuity of a function $f:[a,b]\rightarrow E$ into some Banach space also ensures (i) measurability and (ii) differentiability almost everywhere, like it does on ${\bf R}$?
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20 views

Question on the range of values of compact operator

Suppose $X$ is a Banach space and $T$ is a compact operator from $X$ to $X$. Let $B_1$ is closed unit ball in $X$. Then, can we get a resual that $\overline{T(B_1)} \subseteq T(X)$ ? It is easy to ...
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15 views

Sums of two closed and closed / continuous operators

Let $X$ be a normed space and $A_j:D(A_j)\rightarrow X$ (j=1,2) linear. (i) If $D(A_1)=X$, $A_1$ continuous and A_2 closed. Do we have $A_1+A_2:D(A_1)\cap D(A_2) \rightarrow X$, $x\mapsto A_1x+A_2x$ ...
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1answer
48 views

Showing $(M^\perp)^\perp=\overline{M}$

I have a question about a step in proving $(M^\perp)^\perp=\overline{M}$ where $M$ is a linear subspace of a normed vector space $E$. And $M^\perp=\{f\in E^*|\langle f,x\rangle =0\}$ This is the ...
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17 views

Composition of smooth functions is smooth

Is there any book / online resource where this proof is carried out in the context of Banach spaces and Frechet derivatives?
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1answer
26 views

Differentiation of pointwise composition operator

I'd like to prove that the composition of smooth functions between Banach spaces is smooth. What puzzles me a bit is notation, how do I write the chain rule in terms of functions without explicit ...
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20 views

Clarification of a passage in Leray-Schauder theorem's proof

I'm looking at the proof of Leroy-Scauder theorems's. This is the statements: If $X$ is a Banach space, $K \subset X$ a convex, close and bounded set, $F:K \rightarrow K$ compact then $F$ has a ...
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4answers
50 views

Show that $\ell^1$ is not complete with a certain metric

For $x=(x_j)_{j\in\mathbb N}\in \ell^1$ let $$\|x\|=\sup_{n\in \mathbb N}\left \Vert \sum_{j=1}^{n}x_j\right\Vert$$ Show that $(\ell^1,\|\cdot\|)$ is a normed space, but it is not complete. The ...
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1answer
56 views

Closed spaces in lp and distances

I try solve exercise from Diestel: If Y is a proper closed linear subspace of $l_p$ (1 $<$ p < $\infty$), then there is an x $\in$ $S_x$ so that distance $d(x, Y)$ $>=1$. I try solve this ...
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2answers
39 views

$X+Y$ is closed $\Leftrightarrow$ $\|x\|\leq c\|x+y\|$ for all $x\in X$ and all $y \in Y$.

The problem says Let $(Z,\|\cdot\|)$ be a Banach space. Let $X$ and $Y$ be two closed subspaces of $Z$ such that $X\cap Y=\{0\}$. Prove that $X+Y$ is closed if, and only if, there exists $c\geq 0$ ...
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0answers
36 views

Linear independence of differential 1-forms

Let be $(E,\mathbb{K}),(F,\mathbb{K})$ Banach space and $U\subset E$ open set. If $f_1,...,f_n\in\Omega_1(U;F)$ differential 1-forms are linearly independents, where $$\Omega_1(U;F)=\{f:U\rightarrow ...
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31 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
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1answer
33 views

Every isomorphism on a separable Banach space has a completely invariant dense subset

If $T$ is an isomorphism acting on a separable Banach space, can we always find a countable dense subset $D$ of $X$ such that $T(D)=D? $
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1answer
157 views

Borderline case of interpolation of Banach spaces

Let $B \subset A$ be Banach spaces with a continuous embedding. Is the inequality $$ \|b\|_B \leq C \sup_{t > 0} \inf_{\tilde{b} \in B} \{ \|b - \tilde{b}\|_B + t \|\tilde{b}\|_A \} \quad \forall ...
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21 views

$\ell_q$ is not finitely representable in $\ell_p$ if $2<q<p$.

This seems to be a well known result in Banach space theory. It is referenced, for example, in Pietsch's book "History of Banach spaces and Linear Operator". Where can I find a proof? Who was the ...
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27 views

Unit balls and the Schatten norms

I have a very naive question: Let $A$ and $B$ $n \times n$ (complex) matrices with operator norms $\|A\| \leq 1$ and $\|B\| \leq 1.$ Pick a $1 \leq p < \infty.$ Then with a constant $K_p$ ...
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1answer
39 views

Approximation Property: Decomposition

This is a real question of me. Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional range: ...
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48 views

Distance preserving map on finite dimensional normed space E is a bijection

The problem is to show that for $E$ a finite dimensional normed space and $T:E\to E$ a map (not necessarilly linear) such that $\Vert T(x)-T(y)\Vert =\Vert x-y\Vert \forall x,y\in E$is a bijection. ...
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1answer
31 views

Why isn't the completion of $C^0$ wrt. the $L^2$ norm a space of sequences instead of a space of functions?

We know that $L^2(\Omega)$ can be defined as the completion of $C^0(\Omega)$ with respect to the norm $$\left(\int_\Omega |u|^2\right)^{\frac 12}.$$ But strictly speaking, $L^2(\Omega)$ is a space of ...
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35 views

Differentiation through the integral sign, more general case

I wondered in which cases, given a measurable space $(A,\mu)$, Banach spaces $E,F$, an open $U\subseteq E$ and $f:A\times U\rightarrow F$, we can conclude that the function $s\mapsto\int_A f(x,y)dx$ ...
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1answer
48 views

Approximation Property: Characterization

As reference the german wiki: Approximationseigenschaft Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$ ...
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15 views

Banach Algebra convergence

I am taking a courses on Differential Calculus and I would like to know the details in the steps of the proof (1) & (2). Also I would like some details on how to solve the exercise below. (3) ...
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1answer
37 views

Prove that the space of sequences with limit $0$ is complete.

Prove that $C_0$ (the space of sequences with limit $0$) is complete. My effort: Let {$x_n$} be sequnce in $C_0$ converging to the limit $0$. As the {$0$} is in $C_0$ hence $C_0$ closed in $C$ ...
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1answer
25 views

Strong Topology and Strong Operator Topology on Hilbert Space

Suppose $H$ is a Hilbert space (much of this still works if it's just a Banach space), $x\in H$, and $(x_n)$ a sequence in $H$. Does $x_n\to x$ strongly in H iff $x_n\to x$ as operators in the strong ...
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1answer
34 views

Homeomorphisms of different spheres

It is pretty straightforward to show that if $X$ is a Banach space under two equivalent norms, then the respective open unit balls $B_1$ and $B_2$ are homeomorphic only by showing that $B_i$ is ...
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1answer
48 views

Is any bounded linear operator of dual spaces is dual of a linear operator?

Let $X,Y$ be two Banach spaces and $S:Y^* \to X^*$ be a bounded linear operator. Is there always bounded linear $T: X\to Y$ such that $S=T^*?$
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1answer
68 views

Most general setting for the fundamental theorem of curves

I want to learn more about the fundamental theorem of curves. Wikipedia states the theorem for ${\bf R}^3$ only but I found another source (Theorem 5.5.18, in German only) where it is proved for ...
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31 views

Basic properties of uniform limits in Banach spaces

Where can I find infos (books, keywords, online materials, etc.) about when the uniform limit of a sequence of continuously differentiable functions $f_n:U\subseteq E\rightarrow F$ between arbitrary ...
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35 views

Does every closed, densely operator in a Banach space have an closed, densely defined extension on a Hilbert space?

Assume that $H_1,H_2$ are separable Hilbert spaces, $B$ is a separable Banach space and $H_1\subset B\subset H_2$. Assume further that the inclusion mappings are continuous and have dense images. ...
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49 views

Boundedness of linear operators in $L^p$

I was wondering if the following conditions are equivalent for a linear operator $T$ between $L^p$ spaces: i) T maps $L^p$ to $L^p$, that is, if $f \in L^p$ then $Tf \in L^p$. ii) There exists ...
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1answer
31 views

Equivalent characterizations for Reflexive spaces

Well I'm reading about Reflexive spaces those days and I would like to see a proof for two different claims. The first claim is that a Banach space is reflexive iff every bounded functional attains ...
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21 views

The proof of $(c_0)^* \cong l^1$ always requires construction from $l^1$, not $(c_0)^*$?

Together with the proof that the dual of $l^1$ is $l^\infty$, I understood the element of $l^1$ is the great companion with $l^\infty$, in the sense that $\sum a_nx_n$ absolutely converges, so that: ...
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39 views

Weakly closed $\iff$ closed using the Separation theorem

My question is about the following problem. $X$: Banach space, $C$: convex subsets of $X$. Then, followings are equivalent. i) $C$ is closed. ii) $C$ is weakly closed. I ...
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23 views

isometrically isomorphism [duplicate]

How can embed separable Banach to Cb(X)(family of all bounded continuous functions on topological space X) with non metrizable X ? If X is locally compact or Tychonof is very well. note : We know ...
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1answer
40 views

Is $x\mapsto \| Tx\|$ lower semi-continuous?

Suppose $T:\mathcal D(T)\rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$. Is it true that $$ \|Tx\|\leq \liminf_{n\rightarrow\infty} \|T ...
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1answer
33 views

Bounded Right Inverse

If a linear operator between two Banach spaces is surjective and bounded, can we get any information about a right inverse? For example, is it bounded? Thanks, trying to understand trace operator ...
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0answers
28 views

Showing map is isometry between Banach quotient space

I have a closed subspace $Y$ of a Banach space $X$ and a map $T: X'/Y^{\circ} \to Y'$ given by $[f] \to f|_y$. The norm in $X'/Y^\circ$ is given by $\|[f]\| = \inf \{ \|f-h\| : h \in Y^\circ \}$. I'm ...
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2answers
72 views

Extending linear functional in non-unique way

I'm trying to find an example of when the extension of a functional in the Hahn-Banach theorem is not necessarily unique. I'm looking at the space of continuous functions on $[0,1]$ and I'm trying to ...
5
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1answer
74 views

Showing $\int f(x)e^{nx}$ implies $f(x) = 0$

I'm trying to show that if $f$ is a continuous function on $[0,1]$ and $\int_0^{1} f(x)e^{nx}\,dx = 0$ for all $n = 0,\ 1, 2,\ \dots$, then $f(x) = 0$. I'd like to use the Weierstrass approximation ...
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1answer
77 views

Continuous operator between Banach spaces, closed range

I have some problems proving the following: $T: X \rightarrow Y$ is a continuous, linear operator between Banach spaces. Prove that $T$ is surjective $\iff$ $T^* : Y^* \rightarrow X^*$ is injective ...
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1answer
24 views

Is $Tf=x_0+\int_0^tf(t)(1-f)\,dt$ a Lipschitz function?

Consider the Banach space $C[0,1]$ with the uniform norm and the operator given by $(T(f))x=x_0+\int_0^xf(t)(1-f(t))\,dt$ for $x\in[0,1]$ and $x_0\geq0$. I want to show that there is a number ...
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0answers
40 views

Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
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1answer
10 views

Proving that $(Sf)(x)=1+\int_0^x t^2f(t)\,dt$ is $K$-Lipschitz

I want to show that $S:C[0,1]\to C[0,1]$ where $$(Sf)(x) = 1+\int_{0}^{x} t^2\cdot f(t) \,dt$$ is a Lipschitz map for $x\in[0,1]$. There I use the Euclidean norm on $\mathbb{R}$ and the uniform norm ...
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1answer
50 views

Is there a $f\in C[0,1]$ such that $f(x)=\frac12 \sin (f(1-x))$

Is there a $f\in (C[0,1],\lVert\cdot\rVert_{\infty})$ such that $f(x)=\frac12 \sin (f(1-x))$? I feel like you need to apply Banach´s fixed point theorem, which means it suffices to prove that ...
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1answer
57 views

Completion of a Banach space with respect to a different norm

Let $(X,|\cdot|_X)$ be a Banach space. Define a space $Y$ as the completion of $X$ under a norm $$|u|_Y = |u|_X + |Tu|_Z$$ where $T:X \to Z$ is a linear continuous map where $X \subset Z$ is a ...
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1answer
59 views

A linear bijection to a Banach space must have bounded inverse

Suppose that $X$ and $Y$ are Banach spaces, and $D ⊂ X$ is a linear subspace, which may not be closed. Suppose that $T : D → Y$ has a closed graph (in $X\times Y$), and is $1-1$ and onto. If $D$ is ...
5
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51 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
3
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1answer
31 views

$L^\infty(0,1)$ as $C(K)$

For any $\sigma$-finite measure $\mu$, the space $L^\infty(\mu)$ is isometric (even as a Banach algebra) to the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. For example, ...
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60 views

Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...