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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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 b ...
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Is complemented subspace of complemented subspace is complemented? [on hold]

Let $X\subset Y\subset Z$ be Banach spaces such that $X$ is complemented in $Y$ and $Y$ is complemented in $Z$. Is it true that $X$ is complemented in $Z$?
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$\ell_q$ 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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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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Compact Operators: Decomposition

This is a real question of me. Given a Banach space. Consider a basis on finite dimensional range: $$\dim\mathcal{R}F<\infty:\quad y_1,\ldots, y_N$$ Hahn-Banach lifts the dual basis up: $$ y_n\in ...
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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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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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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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24 views

Is every finite dimensional linear space a banach space [on hold]

Is every finite dimensional linear space a Banach Space? Is every finite dimensional linear space a Hilbert Space?
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57 views

Approximation Property: Hilbert Spaces [on hold]

Note: This thread is not to gain reputation!! Given a Hilbert space. How to prove: It has the approximation property!
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Approximation Property: Characterization

Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_\varepsilon-1\|_C<\varepsilon\quad(T_\varepsilon\in\mathcal{F}(E))$$ Then every compact ...
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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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Prove that $C_c(X)^c=C_0(X)$ if $X$ is locally compact $T_2$ [on hold]

Prove that $C_c(X)^c=C_0(X)$ if $X$ is locally compact $T_2$ So in genarel $C_c(X)$ is not complete wr.t . Supnorm.
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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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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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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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23 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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48 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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20 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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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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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
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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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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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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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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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29 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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20 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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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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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 ...
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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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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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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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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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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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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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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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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 ...
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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 $T_n\in \mathscr B(\mathcal ...
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$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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Semigroups & Generators: Entire Elements: Construction

Problem Given a Banach space $E$. Consider a $\mathcal{C}_0$-group(!): $T:\mathbb{R}\to\mathcal{B}(E)$. Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)$$ (The domain being those ...
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Continuity of the dual product

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 on $X\times X^*$, mainly because ...
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Showing $(\ker(T))^{\circ} = \operatorname{Im}(T')$ in Banach space

For $T: X \to Y$ with $T$ having finite dimensional image I'm trying to show: $$(\ker(T))^{\circ} = \operatorname{Im}(T')$$ Where $T'$ is the dual operator. I've shown that if we take an element in ...
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Locally uniformly convex Banach space, which is not uniformly convex.

I would like to find an example of a reflexive Banach space $X$, which is locally uniformly convex, however, it is not uniformly convex. The motivation is that I am studying degree theory for ...
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Hille-Yosida vs. Lumer-Philips: Reformulation?

I know it is preferable to have more context but let me simply phrase it for now... Considering contraction semigroups: It seems to me that Lumer-Philips is a simple reformulation of Hille-Yosida, ...
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Dual of product isometric to product of dual Banach spaces

I'm tying to show that if $X$ is a real Banach space then there is an isometric injection $$\tau: X' \times X' \to (X \times X)' $$ Where $X \times X$ has the norm $||(x_1, x_2)|| = ||x_1|| + ...
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Example: operator injective, then the adjoint is NOT surjective

Let $T: V \rightarrow W$ be a bounded operator on normed spaces $V,W$. Now, there is a unique adjoint operator $T': W' \rightarrow V'$ defined by $T'(\alpha) = \alpha \circ T$. In finite dimensional ...
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Proving that a Hölder space is a Banach space

I am trying to show that the Hölder space $C^{k,\gamma}(\bar{U})$ is a Banach space. To do this, I successfully proved that the mapping $\| \quad \| : C^{k,\gamma}(\bar{U}) \to [0,\infty)$ is a norm, ...
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How to use Triangle inequality to find the projection onto unit ball?

The projection onto the unit ball $$C:=\mathbb{B}(0,1)=\{x:||x||\leq1\}$$ is given by $$P_{C}(x)=\frac{x}{max\{||x||,1\}}, \quad\forall x\in X$$ where $X$ is Hilbert space. Now I can understand this ...
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Is there a non-dense subspace of $\ell_1$ which is dense in every finite-dimensional subspace?

We want to find a subspace $F \subseteq \ell_1(\mathbb{R})$ such that for any finite $n \in \mathbb{N}$ and tuple $x=(x_1,\dots,x_n) \in \mathbb{R}^n$ we can find a sequence $(f^k)_k \subseteq F$ with ...
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Explicit Hahn-Banach extension formula in finite dimensional $l^p$ /Smoothness of the Hahn-Banach mapping

Consider the finite dimensional vector space $V=(\mathbb{R}^N,\|\cdot\|_{p})$, equipped with the usual $l^p$ norm, $1<p<\infty$. Consider a linear subspace $U\subset V$ (not necessarily a ...