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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weak$^∗$ neighborhood of $x$ in $\ell_1$

I have this problem Let $x \in \ell_1$ and $\epsilon>0.$ Choose an $N\in N$ such that $\sum\limits_{k=N}^{\infty}|x_k|<\epsilon$ I cannot understand why V is a weak$^∗$ neighborhood of $x$ in ...
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28 views

Frechet derivative and Gateaux derivative

Smulian lemma says Let $(X, ||.||)$ be a Banach space and$(X^*, ||.||^*)$ and let $x\in S_X=\{x\in X:||x||=1\}$ then (i) $||.||$ is Frechet diffrentiable at $x$ iff ...
-1
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1answer
62 views

Example of open operator but not closed [closed]

Assume that $T:\ell_1\to\ell_2 $ is bounded,linear and one-to-one. Prove that $T(\ell_1)$ is not closed in $\ell_2$
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1answer
55 views

the point where all functional are non zero

Let $\{f_n\}$ be sequence of non zero bounded linear functionals on a Banach space X. Show that there is $x\in X$ so that $f_n(x)\ne0$, for all $n\in \Bbb N$. I am confused, non zero functional ...
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0answers
24 views

A question about uniform convergence in proof that $L^\infty$ is Banach.

I was reading this post Understanding proof of completeness of $L^{\infty}$ and it is mentioned that the sequence $(f_n)_{n\in\mathbb N}$ converges uniformly on a conegligible set $N^C$. Could someone ...
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1answer
34 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
2
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1answer
40 views

Riesz Lemma for reflexive spaces

I know the proof of Riesz Lemma: Let $Y$ be a closed (proper) subspace of a normed space $X$. Let $\varepsilon >0$. Then it exists an element $x \in X$ such that $||x||=1$ and $d(x, Y) \geq ...
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1answer
58 views

An example of a separable Banach sequence space in which the finite support sequences are not dense?

I am wondering if there exist examples of Banach (or Frechet) sequence spaces in which the set of all finite support sequences are NOT dense?
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0answers
29 views

Weakly compact closed balls in reflexive space

If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not ...
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1answer
19 views

Extension of a linear operator

Let $T$ be a linear operator defined on the space of the algebraic polinomials in $[0,1]$ (polinomials with rational coefficients) such that for each $k \in \mathbb{N}, T[x^k]=0$. Is it possibile to ...
0
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1answer
41 views

Conicide of $w^*$ and norm topology on $S_{\ell_1}$

I want to show that on $S_{\ell_1}=\{x\in \ell_1: ||x||=1\}$the $w^*$-and the norm topologies are coincide. Can any one help me . Thanks
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4answers
148 views

Topological Vector Space: $\dim Z\text{ finite}\implies Z\text{ closed}$

Let $V$ be a Hausdorff topological vector space and $Z$ a linear subspace: $Z\leq X$ Is there a neat way to prove that: $$\dim Z\text{ finite}\implies Z\text{ closed}$$
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2answers
29 views

How to show $\{f_n\}_{n=1}^\infty$ has uniformly convergent subsequence on [0,1]?

Let $\{f_n\}_{n=1}^\infty$ a sequence of second order differentiable functions on the interval [0,1]. If $\forall n\in \Bbb N$ $f_n(0)=f_n'(0)=0$ and for all $n\in \Bbb N$ and $x \in [0,1]$ , ...
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0answers
31 views

Reference/confirmation of a result in analysis

Does anyone know of or have a reference for the following result: Let $X$ be a reflexive Banach space with dual $X^{*}$. If there exists a continuous mapping $f: K \rightarrow X^{*}$ on compact ...
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1answer
32 views

Isometric isomorphism between $\ell^{1}(\mathbb{R}^{3})$ and $\ell^{\infty}(\mathbb{R}^{3})$

Why is there no isometric isomorphism between $\ell^{1}(\mathbb{R}^{3})$ and $\ell^{\infty}(\mathbb{R}^{3})$? I know that there is such an isomorphism if $\mathbb{R}^{3}$ is replaced with ...
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0answers
105 views

Asymptotically isometric copy of $\ell_1$

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that ...
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0answers
27 views

Equivalent Frechet differentiable norm on $\ell_1$ and $c_0$

Does there exist an equivalent Frechet differentiable norm on $\ell_1$ and $c_0$? I think we can not find an equivalent norm on $\ell_1$ but we could find an equivalent norm on $c_0$, I do not prove ...
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1answer
43 views

Frechet differentiable implies reflexive?

Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive? Can any one help me? thanks
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0answers
36 views

Riemann Sphere: Holomorphic Functional Calculus

Why do we consider the holomorphic functional calculus on the Riemann sphere rather than the complex plane only? Is there a serious problem? Moreover isn't any curve encircling the spectrum ones ...
0
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1answer
14 views

I need help showing something is a linear continuous operator.

Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to ...
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0answers
35 views

Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in ...
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0answers
45 views

Applications of uniformly convex and uniformly smooth of Banach space

I am studying on geometry of Banach space, I want know applications of uniformly convex and uniformly smooth of Banach space in some branches of mathematics and engineering. Can you help me Thanks ...
2
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0answers
41 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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1answer
30 views

The modulus of smoothness of $c_0$ by an equivalent norm

Let $(X,\|\cdot\|)$ be a Banach space. For $t>0$, the modulus of smoothness of $\|\cdot\|$ is defined by $\rho_X(t)=\sup\left\{\dfrac{\|x+ty\|+\|x−ty\|}{2}−1:x,y\in S_X\right\}$. We define an ...
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1answer
108 views

Continuity of $|.|$ in $W^{1,p}_0$

please i dont understand this proof We suppose that $u\rightarrow u$ on $W^{1,p}_0$ i dont understand why we must use the weak compactness and the uniform convexity of $W^{1,p}_0$ ? Thank you
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0answers
21 views

Banach-valued holomorphic functions [duplicate]

Let $X$ be a Banach space. Can we define holomorphic functions $f:\mathbb{C}\to X$ by the notion of derivability i.e. $$\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}$$ Do we still have equivalence between ...
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1answer
63 views

“right shift” il $L^1$

Let $X=L^1(\mathbb{R})$ be the space of Lebesgue integrable functions $f:\mathbb{R}\rightarrow \mathbb{C}$ with the usual norm. Let $T\in B(X)$ be defined by $$(Tf)(t)= f(t+1)$$ I need to find the ...
2
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1answer
43 views

Prove that this space is not Banach

Let $\Omega\subset\mathbb{R}^n$ be an open, bounded set with boundary $\partial\Omega$ of class $C^1$. $$\mathcal{A}:=\{u\in C^2(\bar\Omega):u=0\text{ on }\partial\Omega \}$$ endowed with the scalar ...
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0answers
30 views

Group of operators such that $|T(t)x|\geq c |x|$

Let $X$ be a Banach space. Can I have an example of a strongly continuous group of operators $T(t)$ such that $$|T(t)x|\geq c |x|, \ t\in\mathbb{R}$$with $c>1$. For $c=1$, I know examples of ...
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1answer
26 views

Verification of conclusions regarding duality maps

I have two conclusions drawn from two results. I want to know how valid these two conclusions are. Firstly Consider the duality mapping(set-valued) $J:X \rightrightarrows X^{*}$ defined: $J(u) := \{ ...
2
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1answer
49 views

Linear Functional: Continuous? [duplicate]

Given a Banach space: $E$ and chosen a Hamel basis: $\mathcal{B}$ Any vector induces a (noncanonical) algebraic linear functional by: $$\delta:E\to E^*:\delta_b(b'):=\delta_{b,b'}\text{ defined ...
2
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1answer
51 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
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1answer
33 views

$\sup_t |T(t)|<+\infty$ implies $\sup_t |T(t)^*|<+\infty$?

Let $X$ be a Banach space. $T(t)$ a family of bounded operators for $t\in\mathbb{R}$. $T(t)^*$ is the adjoint operator of $T(t)$. If $\sup_t |T(t)^*|<+\infty$ , then by Hahn-Banach, there's a ...
2
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1answer
33 views

uniform continuity of the function $t\mapsto\langle x^*,f(t)\rangle$

Let $X$ be a Banach space. $f:\mathbb{R}\to X$ a function. If we have $t\mapsto\langle x^*,f(t)\rangle$ uniformly continuous on $\mathbb{R}$ for each $x^*\in D$ where $D$ is a dense subset of $X^*$ ...
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22 views

Complexity of a Borel linear subspace of a Banach space

This question is inspired by the MO question Image of $L^1$ under the Fourier transform, but I think it might be much easier so I am posting it here instead. Let $(X, \|\cdot\|)$ be a separable ...
1
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1answer
52 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
0
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1answer
26 views

Simple question about convergence and Gateaux derivative

If I consider the sequence $\{x_n\}\in L^2(\Omega)$ such that: $$ x_n \rightarrow x $$ We know that $x\in L^2(\Omega)$ because we're in a Banach space. So I can say that ...
5
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1answer
137 views

If $P$ has marginals $P_1, P_2$, is $L^1(P_1) + L^1(P_2)$ closed in $L^1(P)$?

Suppose that $\mathbb{X}=\mathbb{X}_1\times \mathbb{X}_2$ and suppose that $ P$ is a probability measure on $\mathbb{X}$ with marginals $ P_i$ on $\mathbb{X}_i, i=1,2$, i.e., $$\int f_i(x_i)\, ...
1
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1answer
29 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt ...
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0answers
24 views

Dual space of $L^2(0,T;H^1) + L^p(0,T;L^p)$ and its duality pairing?

Let $V=L^2(0,T;H^1) + L^p(0,T;L^p)$. We know that its dual space is $V^* = L^2(0,T;H^{-1}) \cap L^p(0,T;L^p)$. So if $v \in V$, then by definition $v=a+b$ where $a \in L^2(0,T;H^1)$ and $b \in ...
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2answers
35 views

Fredholm Index: Finite Corank $\Rightarrow$Closed Range [duplicate]

Obviously closed subspaces turn quotient spaces into normed spaces rather than just merely vector spaces. However the dimension involved in Freholm's index are purely algebraic. Why do we thus ...
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1answer
78 views

What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...
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1answer
51 views

Compact Operator <=> Separable Range

Is it true that a bounded operator is compact iff its range is separable: $$T\text{ bounded}:\quad T\text{ compact}\iff \mathcal{R}(T)\text{ separable}$$
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1answer
34 views

Existence of nontrivial bounded linear operator?

Are there two normed spaces such that there is no nontrivial bounded linear operator between them: $$\nexists T:X\to Y: T\text{ nontrivial, linear and bounded}$$
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1answer
27 views

Inequality norms

Let $A$ be a bounded linear operator on a Banach space $X$. Can we show that for an arbitrary $n \in \mathbb{N}$ and $x \in X$ such that $\|x\|_X \geq 1$ we have that $$\|A^n x \| \leq \|Ax\|^n.$$ ...
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1answer
23 views

Convergence of the sequence of operators on a Banach space.

Let $T$ be a bounded linear operator on a Banach space $E$, and let $(T_n)$ be a sequence of bounded linear operators on $E$ such that $$\| (T - T_n)x \| \to 0 \ \ \text{ as } n \to \infty$$ for all ...
2
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2answers
102 views

Resolvent Set: Definition

Given Banach spaces: $X,Y$ Consider a linear operator: $T:\mathcal{D}(T)\to Y$ (not necessarily bounded nor closed nor closable nor densely defined) Define for the shorthand the shifted operator: ...
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28 views

Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.

The Assignment: Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the continuous function $f$ for all $n \in \mathbb{N}$: $$f_n: [0,1] \rightarrow \mathbb{R}, \ ...
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1answer
69 views

What is the dual space in the strong operator topology?

Let $X$ be a Banach space, the strong operator topology on the space of bounded linear operators $\mathcal{B}(X)$ is defined by the family of continuous semi-norms $A\to\|Ax\|$, $x\in X$. What is the ...
2
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1answer
82 views

Have they solved this exercise correct?(banach space, function space).

Please look at this exercise: It is the last question I have a problem with Here is the solution: They say that $\|I(1)\|=\sup|g(t)|$. But isn't $\|I(1)\|=\int_0^1g(t)dt$? If so, what is the ...