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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How to detect reflexivity of the closure

Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue $L_p$ norm is reflexive when $1 < p < \infty$, but it is not reflexive for $p = 1$. How ...
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71 views

A homogeneous but slightly asymmetric inequality involving $L_1,L_{p-1}$ and $L_p$ norms.

I need to prove or disprove the following inequality: for any $Z=(z_1,\ldots,z_l)\in\mathbb{C}^L$ and for any $p\geq 2$, $$\biggl|\biggl\|\sum_{j=1}^L z_j\biggr\|^p - ...
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40 views

Prove that this operator is continuous [duplicate]

Let $X,Y,Z$ be Banach spaces, and let $T:X\to Y$ be linear. Let $J:Y\to Z$ be linear, bounded and injective. If $JT:X\to Z$ is bounded, then T is bounded.
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Weird definition of norm's triangle inequality

Cassels (Froehlich & Cassels, ANT) uses a rather unusual definition of triangle inequality when he defines a norm. He states that a norm should satisfy the following (in lieu of ...
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45 views

A reflexive subspace is weak* closed in the bidual.

Let $X$ be a Banach space. Let $Y$ be a closed subspace. I am working on an exercise for which the hint reads: If $Y$ is reflexive, $Y$ is weak* closed in $X^{**}$. Can anybody explain why the ...
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Are these subspaces of continuous, bounded functions Banach Spaces?

Let $X=C^b(\mathbb{R})$ be the space of continuous, bounded functions in $\mathbb{R}$. $X$, equipped with the norm $\|f\|_\infty=\sup_{x\in\mathbb{R}}|f(x)|$ is a Banach space. What can you say about ...
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146 views

Hahn-Banach via Hamel Basis

my question for tonight: Is there a proof for Hahn-Banach using a Hamel Basis? I know, the proof for existence of Hamel Bases uses already Axiom of Choice, but I'd like to apply this without refering ...
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1answer
89 views

A Banach space is reflexive if a closed subspace and its quotient space are both reflexive

Let $X$ be a Banach space. Let $Y$ be a closed subspace. Suppose that the normed spaces (in fact Banach spaces) $Y$ and $X/Y$ are both reflexive. I need to show that $X$ is reflexive. I cannot show ...
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37 views

Showing a map is continuous in Banach space

We have maps $F:X \times X \to \mathbb{R}$ and $f:X \to \mathbb{R}$ on a Banach space $X$. Let $F(u,v) = f(u+v) - f(u-v)$. The map $f$ is nonlinear and not identically zero (to avoid trivial ...
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19 views

A boundedness condition on Banach space

We have maps $F:X \times X \to \mathbb{R}$ and $f:X \to \mathbb{R}$ on a Banach space $X$. Let $F(u,v) = f(u+v) - f(u-v)$. The map $f$ is nonlinear and not identically zero (to avoid trivial ...
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45 views

Is this a separable Banach space?

Define $X=\{ f\in C^1 ( \mathbb{R} ):~\int_{-\infty}^{\infty} \left| f'(x) \right| \mbox{d}x < \infty \}$ with norm $\| f \| = \left| f(0)\right| + \int_{-\infty}^{\infty} \left| f'(x) \right| ...
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1answer
28 views

Is best approximation from a linear subspace a linear map?

Let $X$ be a strictly convex Banach space, and $Y \subset X$ a closed subspace. Then for any $x \in X$ there exists a unique $y \in Y$ that minimizes the distance to $x$, i.e. a best approximation of ...
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2answers
86 views

Collecting things that are preserved by (isometric) isomorphisms between normed spaces

I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional ...
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1answer
51 views

Is the dual of a reflexive Banach space strictly convex?

Is the dual of a reflexive Banach space strictly convex? Why? This is a question that arouse trying to understand the theory behind approximation by finite element methods.
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39 views

Easy question about $C_c^\infty(0,T)$ and $C_c^\infty((0,T);X)$

Let $f \in C_c^\infty(0,T).$ It follows that $f \in C^k(0,T)$ for all $k$, and so if $t_n \to t$ then $$|f(t_n) - f(t)| + |f'(t_n) - f'(t)| + ... +|f^{(k)}(t_n) - f^{(k)}(t)| \to 0$$ for all $k$. Now ...
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2answers
84 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
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36 views

properties of integral operator $x^{-1}\int_0^xf(x,y)v(y)dy $

here we have two cases to study $(1)$ let us fix any $f \in C^{1}[ [0,1] \times [0,1]]$ ($k \neq 0$). Set $$[T(v)](x) := x^{-1}\int_0^xf(x,y)v(y)dy $$ for any $x \neq 0$ otherwise $[T(v)](0) := ...
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1answer
43 views

Is the canonical injection between $ C^{k+1}[0, 1] $ and $ C^k [0, 1] $ compact?

If $ k=0 $ by the Ascoli - Arzelá theorem the answer is yes, but i don't know how to proceed in the general case ($ k> 0, k \in \mathbb{N} $). I tried to build a counter example using Riesz lemma ...
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34 views

Proof of an equivalence in Hilbert spaces

Let $H$ be a Hilbert space. Prove that the following are equivalent: a) the algebraic dimension of $H$ is finite; b) each closed, not empty subset $C$ has an element of minimum norm (that is the ...
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64 views

Sequential version of the Eberlein-Shmul'yan theorem

Theorem: A Banach space is $(i)$ reflexive iff $(ii)$ every bounded sequence possesses a weakly convergent subsequence; see e.g. Thm 3.18 and 3.19 in Brezis' 2010 book. The implication $(i) \implies ...
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49 views

Relationship between a finite codimensional subspace of dual space and the annihilator

Notation: $X$ is a banach space, $X'$ is the dual space to $X$. When $V \subset X'$, we write $\ker V = \cap_{l \in V} \ker l$, and when $W \subset X$, we write $ann \; W = \{l \in X' \mid l(w) = 0 ...
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33 views

What is $\|a\|=\max_{f \in A^\ast}|f(a)|$?

I would like to ask if the norm $$\|a\|=\max_{f \in A^\ast}|f(a)|$$ has a name (where $A$ is a Banach space and $A^\ast$ denotes the continuous dual) and how to prove this equality. This equality is ...
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property of a given subset $ A $ of $\mathit{l}^2 $

I was given this exercise Let $\lbrace e_n \rbrace $ be the canonical basis of $\mathit {l}^2 $ (the space of square sommable series with the usual norm) and set $$ A:= \lbrace \sum_{n \ in ...
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1answer
15 views

Verify that a Weak-neighborhood of a point is contained in the kernel of a continuous linear functional

I've encountered several times this reasoning but i can't find a good answer to it. Let $ E, F$ be banach spaces Given $\varphi : E \to F $ linear and continuous w.r.t the weak topology $\sigma (E, ...
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40 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
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1answer
34 views

differences between Banach spaces and $\Bbb R^n$.

Can you please tell me, what are the biggest differences between Banach spaces and $\Bbb R ^n$? I am trying to understand the Frechet derivative.
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clarification about theorem $3.10 $ of Brezis functional analysis book.

I'm referring to the theorem at page $61$. It shows that for a linear operator $T $ between $E $ and $ F$ Banach space are equivalent (notation: $S$ means strong topology, the norm ome, $W $ weak ...
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1answer
29 views

Constant function on Banach space [closed]

Let $f:X\longrightarrow \mathbb{R}$ be the function of $C^{1}$ class ($X$ - Banach space). Show that if $$f'(x)(x)=0\quad \textrm{for all} \quad x\in X,$$ then $f$ is constant.
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Function on $ l_{\mathbb{R}}^{\infty}$ [duplicate]

For which points function $$f:l_{\mathbb{R}}^{\infty}\ni x\longrightarrow ||x||_{\infty}\in \mathbb{R}$$ is differentiable?
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1answer
60 views

Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
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56 views

Hyperplane which can separate two closed convex set

Define: $$E=:\ell^1(R)=\{x=(x_n)_n: \|x\|_1=\sum_{n=1}^{\infty}|x_n|<\infty\}$$ We have two closed convex sets $X,Y$ as subsets of normed vector space $\ell^1(R)$ with $X\cap Y=\emptyset$ and $Y$ ...
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1answer
45 views

Check in which points is f differentiable.

Check in which points is f differentiable: $f: l_{ \Bbb R}^{ \infty} \ni x \rightarrow ||x||_{ \infty} \in \Bbb R$ I started with this: $ \lim_{||h||_{\infty} \to 0} ...
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118 views

Isomorphisms between Normed Spaces

Between two normed (linear) spaces there are several notions of isomorphisms: Linear isomorphisms: linear bijective maps Topological isomorphisms: linear homeomorphisms (due to the linearity these ...
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2answers
116 views

Differences between $L^p$ and $\ell^p$ spaces

Could someone explain some differences between the $L^p$ and $\ell^p$ spaces? Thanks a lot.
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46 views

About the closedness of Banach space

I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I ...
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1answer
68 views

Prove that the Set of Bounded Linear Operators is Banach

Let $B(V,V')$ be the vector space formed by set of linear operators $T:V\rightarrow V'$. where $V,V'$ are normed vector spaces. Equip $B(V,V')$ with the norm $$ \|T\|=\sup\frac{\|T(x)\|}{\|x\|} $$ ...
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113 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to ...
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1answer
114 views

A contraction approach on Collatz conjecture [closed]

Talking about Collatz like iterative sequences, If we define Collatz function as $f(x)=k(2.5x+1)+x/2$ where $k=x$%2 then according to Banach fixed point theorem(from wiki), "Every contraction ...
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Reference request for the fact

Does anyone know a reference to the paper or a textbook where this fact is proved $$ \mathcal{B}(\bigoplus_1 X_\alpha, Y)\cong_1 \bigoplus_\infty \mathcal{B}(X_\alpha, Y) $$ Most author are bored to ...
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$\ell^p$ is not isometric to $\ell^q$

The problem is this: if $1\le p<q<\infty$ then $\ell^p$ and $\ell^q$ are not isometric (as Banach spaces). This is an exercise but I'd like to see an elegant proof.
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61 views

$1$-subsymmetric basis

Let $X$ be a Banach space with a $1$-subsymmetric basis $(e_i)$. I'm trying to understand why it is the case that for any $x = \sum_{i=1}^\infty x_i e_i \in X$, any strictly increasing sequence $(n_i) ...
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1answer
82 views

wanted: a lucid demonstration that these $m+n$ bilinear equations have a solution

I shall first state the problem, and if it arouses any interest, I may gradually add a few notes, to provide a little context for those who, like myself, appreciate such background information. let me ...
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40 views

Prove a set is closed using continuous function

Suppose $U$ is an isometry isomorphism from $C(Q)$ to $C(K)$ where $C(Q)$ is the Banach space which contains all real value continuous function defined on $Q$ and its norm is the sup-norm, i.e. ...
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Continuous Linear Functional on $\ell^{\infty}$

I'd like help answering two questions. 1) Prove that there is a continuous linear functional on $\ell^\infty$ such that $f(e_n)=0 \ \forall n \in \Bbb{N}$ and $f(a)=5$ where $a=(1,1,1,1,1,1,\ldots)$. ...
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what is a “Banach algebra” without the norm condition on a continuous multiplication?

I wish to use the following finite-dimensional Banach spaces. although they do not need to be Banach algebras for my proposed application, a mild curiosity is aroused, because the multiplication ...
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25 views

Show that the imbedding $C^{m+1}(\overline{\Omega}) \to C^{m,1}(\overline{\Omega})$ is not compact

Let $\Omega \subseteq \mathbb{R}^n$ be open. Let $C^m (\overline{\Omega})$ be the Banach space of functions such that each partial derivative $D^{\alpha}f$, $|\alpha| \le m,$ exists and is uniformly ...
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45 views

Can a vector space be complete for two non-compatible norms?

Let $V$ be a vector space and suppose that we have two non-compatible norms on it, i.e. I distinguish $E = (V, \|\cdot\|_1)$ from $F = (V, \|\cdot\|_2)$ and I ask that $\not\exists C>0 \; ...
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1answer
59 views

An Example Of A Banach space. [closed]

Consider the linear space $\mathcal{L}_\infty$ and let $x\in\mathcal{L}_\infty$ where $x=(a_1,a_2,...,a_n)$ and taking the norm of $x$ to be $\sup x_i$. My questions are: 1). When defining $x$ in ...
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50 views

BV is Banach space. [closed]

I've found already a question about the demonstration that the space $BV([a,b])$ is a Banach Space, but I've a (probably silly) question... In that demonstration there is that: $$ ...
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1answer
38 views

Show that the following functional is Frechet differentiable in Hilbert space

I need to show that the following functional is Frechet differentiable: $$ f(u) = \|u\|^2_{H} \ \ \text{in a real Hilbert space} \ \ H $$ Solution: As far as I understand, I need to take a Taylor ...