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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Application of uniform boundedness principle for sequence of operators

Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators. Suppose: $T_n$ is bounded $(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in ...
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14 views

Infinite dimensional $\sigma$-compact space

It is well-known that Banach space is $\sigma$-compact iff it is finite dimensional. I'm looking for examples of infinite dimensional normed $\sigma$-compact spaces.
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34 views

Weak convergence in Banach space

If $X$ is a Banach space, $X'$ its dual and $x,x_n\in X$ and $x',x_n'\in X'$, then the following implication holds: $x_n\to_w x$ (weak convergence) and $x_n'\to x'$ in $X'$ $\implies$ ...
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15 views

Space of polynomials as a continuous image of F-space

Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ? My ...
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22 views

Inequality in Banach space [duplicate]

So I have to either prove or disprove this inequality: $$ \left\lVert x\right\rVert^2 - \left\lVert y\right\rVert^2 \le \left\lVert x-y\right\rVert \left\lVert x+y\right\rVert$$ I know this to be ...
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36 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
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12 views

infimum of operator norms of iterations of linear operators

I am currently reading a proof in which a fact is used without proof: For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n ...
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118 views

Weak topology and strong topology in a Banach space.

I have a doubt about weak topology in a Banach space. Let $\mathcal{B}$ a infinite dimensional Banach space, I understood that the weak topology in $\mathcal{B}$, is the topology generated by $\Sigma ...
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33 views

Subspace of a weakly sequentially complete is weakly sequentially complete

A Banach space $X$ is called weakly sequentially complete if all weakly Cauchy sequences are weakly convergent. Question: If $Y$ is a subspace of a Banach space $X$, must $Y$ be weakly sequentially ...
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39 views

To show a map is an isomorphism

In the proof of Lemma $15$, the author claimed that if there exists constants $C>0$ and $D>0$ such that $$C \sup \{ | \sum_{i=1}^k{\alpha(i) f(x_i) | : \| f \|_{\infty}^1 \leq 1, f(0)=0} \} ...
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36 views

Spectral radius and dense subspace

Let $X$ be a Banach space, and let $E$ be a dense subspace of $X$. Let $A: X \to X$ be a bounded operator on $X$ that maps $E$ to itself. Assume that the spectral radius of $A$ restricted to $E$ is ...
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31 views

Showing that $(G' \circ u)u' \in L^p(I)$ where $G \in C^1(\mathbb{R})$ and $u \in W^{1,p}(I)$

I am trying to prove the rule of differentiation of a composition for weak derivatives in Sobolev spaces following the proof given in Corolary 8.11 in Functional Analysis, Sobolev Spaces and Partial ...
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1answer
21 views

Why $Y=\{ f \in C^1([0,1]^n) : f(0)=0 \}$ is a closed subspace of codimension $1$

Suppose $C^1([0,1]^n)$ is the set of real-valued functions defined on $[0,1]^n$, whose derivative $\leq 1$ is continuous on $[0,1]^n$. Define $$Y=\{ f \in C^1([0,1]^n) : f(0)=0 \}$$ Why $Y$ is a ...
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17 views

Meaning of standard integral convolution

In this paper, in the proof of Lemma $13$, there is this sentence: Now, we find a $1$ Lipschitz $\bar{g} \in C^1(\mathbb{R}^n)$ with $\| f - \bar{g}\|_{|\infty} < \epsilon / 2K$, using the ...
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39 views

banach space bigger than $L^p$

we know that $L^p$ is banach space for any $p\geq 1$. My question: Is there any other banach space that is bigger than $L^p$?. In fact, I have an exercice that I don't have any idea: prove that ...
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1answer
32 views

Multilinear Mappings

Let $E$, $F$ complex Banach spaces and $p,q\in \mathbb{N}$ with $p+q\geq 1$. I will denote by $\mathcal{L}_a(^{p,q}E;F)$ the subspace of all $(p+q)$-linear mappings $A\in ...
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23 views

‎‎‎$‎‎C^*$-algebra generated by ‎$‎‎a$‎

Let ‎$‎‎A$ ‎be a unital ‎‎‎$‎‎C^*$-algebra. ‎‎ Assume that ‎$‎‎a\in A$ ‎is a ‎‎normal ‎and ‎invertible element ‎i.e ‎‎$‎‎aa^*=a^*a$ ‎and ‎‎$‎‎aa^{-1}=a^{-1}a=1$‎.‎ ‎let $‎‎C^*({a}) $ be the ...
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28 views

Banach Tarski proof understanding

Theorem (Banach-Tarski Paradox): The unit ball $\mathbb{D}^3 \subset \mathbb{R}^3$ is equi-decomposable to the union of two unit balls. Proof: Let $\mathbb D^3$ be centered at the origin, and $D^3$ ...
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2answers
33 views

Reflexive Banach spaces, compactness

Let $X$ be a reflexive Banach space. Then, consider a linear and compact operator $T \colon X \to X$. Prove that if: $\text{inf} \{ \|Tx\| : x \in X\quad \text{s.t.}\quad \|x\| = 1 \} > 0$, ...
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30 views

Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
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38 views

Corresponding norm from a dual norm?

Let $(X,N_1)$ be a Banach space (separable if necessary) and let $(X^*,N_1^*)$ be its dual space. Here $N_1^*$ denotes the classical dual norm associated to $N_1$. Let $N_2^*$ be an equivalent norm ...
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37 views

$C ([1,2] \times [0,1] \to \mathbb R)$ dense in $C ( [1,2] \rightarrow L^{2} ([0,1] \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
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1answer
57 views

Pythagorean theorem does require the Cauchy-Schwarz inequality?

In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved ...
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145 views

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$.

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$. Normed spaces are equivalent if there are bounded linear maps which are inverses of ...
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31 views

Looking for an (outside $\Bbb R$) application of a certain theorem

I have the following theorem in the lecture notes: Let $E$ be a normed vector space and $\Omega \subset E$ be open and connected, and let $F$ be a Banach space. Let $(f_n)$ be a sequence of ...
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26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...
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20 views

What is the standard notation for the Banach space of functions that vanish at infinity?

Given a topological space $X$ and a real (or complex) valued function $f$ on $X$, we say that $f$ vanishes at infinity if for any $\varepsilon>0$ there is a compact $K \subseteq X$ such that ...
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21 views

Check proof that the embedding of the unit ball $B\subset X$ into $X^{**}$ in weak-* dense

I have to prove the following theorem: Let $X$ be a (real) Banach space, and let $B$ denote its closed unit ball, and let $\tau (B)$ denote its canonical embedding into $B^{**}$, the closed unit ...
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1answer
29 views

Proof on Differentiation in Banach spaces

Prove that f: $\Bbb R^2$-> $\Bbb R$, (x,y)$\mapsto$ x$^2$+ 2xy$^2$ +5y$^3$ is differentiable at (2,1) with DF(2,1)=[6,3]. Now I know that the partial derivatives 1) $\partial f/\partial x (2,1)=2x + ...
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1answer
26 views

Find Riesz representation of $\phi=f({1\over 2})$

"Let $\rho$ be a space of complex polynomial and define $<f,g>={1\over 2\pi}\int_{0}^{2\pi}f(e^{it})\overline{g(e^{it})}dt$ for $f,g:\rho\to \Bbb{C}$. Let $\phi$ be a linear functional on ...
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37 views

$L_2\cap L_\infty$ Banach space - proof check

Problem: $$X=L_2((1,\infty))\cap L_\infty((1,\infty))$$ $$\|x\|=\max\{\|x\|_2,\|x\|_\infty\}$$ Show $(X,\|\cdot\|)$ is Banach space. Attempt: In Banach space every cauchy sequence converges: ...
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Brezis Exercise 3.27 extension.

Let $E$ be a separable Banach space with norm $\|\cdot\|$. The dual norm on $E^*$ is also denoted by $\|\cdot\|$. Let $(a_n) \subset B_E$ be a dense subset of $B_E$ with respect to the strong ...
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15 views

inclusion $C^r(U,C^r(\mathbb{R}^m,Y)\subset C^r(U\times\mathbb{R}^m,Y)$

Let $U\subset\mathbb{R}^n$ be open, $Y$ Banach space, $r\in\mathbb{N}$. Define a map $C^r(U,C^r(\mathbb{R}^m,Y)\to C^r(U\times\mathbb{R}^m,Y)$ by $f\mapsto ((x,y)\mapsto f(x)(y))$ Question: Is ...
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1answer
45 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
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2answers
177 views

Exists a uniformly convex norm on Banach space satisfying certain condition?

Let $E$ be a Banach space with norm $\|\cdot\|$. Assume that there exists on $E$ an equivalent norm, denoted by $|\cdot|$, that is uniformly convex. Given any $k > 1$, does there exist a uniformly ...
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1answer
17 views

Bounding the total variation for Banach lattice

Let $E$ be a Banach lattice and $E_+$ denote its positive cone. For $x_1, \ldots, x_n \in E_+$, is there any way to bound $\sum_1^n\|x_i\|$ with $\|\sum_1^n x_i\|$ without using n? Similaryly, for ...
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39 views

Sum of Banach Spaces is complete

Let $A_1, A_2,...$ be a sequence of Banach spaces with $\|\cdot\|_n$ denoting the norm on $A_n$. Let $p\in[1,\infty)$ and $$\sum\limits_pA_n:=\{(a_n)_{n=1}^{\infty}| a_n\in A_n \text{ and } ...
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1answer
25 views

Countable Complete Orthonormal Set implies countable dense subet

Let $\mathcal H$ be a Hilbert Space, let $B = \{u_j\}_{j=1}^{\infty}$ be a countable orthonormal basis. So we know that if a set is a complete orthonormal basis, the set of all finite linear ...
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Cesaro and Tandori sequence spaces, representations and duality

Definitions. Fix $1\leq p\leq\infty$. Given a scalar sequence $a=(a_n)_{n=1}^\infty$, denote by $\tilde{a}=(\tilde{a}_n)_{n=1}^\infty$, where each $\tilde{a}_n=\sup_{k\geq n}|a_k|$. Now we define ...
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3answers
67 views

Is $C^1([0, 1], E)$ dense in $C([0, 1], E)$ for a general Banach-space $E$?

I'm wondering if there is a result stating that the space of differentiable functions $C^1([0, 1], E)$ is dense in $C([0, 1], E)$ for a general Banach-space $E$. We define the derivative of a function ...
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1answer
75 views

Subspaces of $C^\alpha [0,1]$ are finite dimensional if closed in $C[0,1]$

For $0 < \alpha < 1$, let $C^\alpha([0,1])$ be the subspace of $C[0,1]$ consisting of continuous functions with norm $$ \| f\|_\alpha = \|f\| + \sup_{x\neq y} \frac{|f(x) - ...
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1answer
35 views

A normed space is Banach iff its unit sphere is complete [duplicate]

Let $X$ be a non-trivial (other than singleton $x$) normed space. Prove that $X$ is a Banach space if and only if $\{x \in X \mid \|x\| = 1 \}$ is complete.
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1answer
28 views

Inverse bounded in a Banach space.

Let $X$ be a Banach space and let $A: X \rightarrow X $ be a bounded linear operator such that $A'(\tilde{X})=\tilde{X}$, show that $A$ has a bounded inverse (on its range). If someone could proof ...
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1answer
48 views

Spectrum $\sigma(T)$ of $T:l^1 \to l^1$ given by $T((a_j))=\left( \sum_{j=2}^{\infty} a_j \right) e_1 + \sum_{j=2}^{\infty} a_{j-1} e_j$

I'm considering the bounded linear operator $T$ on $l^1$ (the space of all absolutely convergent complex sequences) given by (with $e_k=(\delta_{kj})_{j=1,2,...}$) $$T((a_j))=\left( ...
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17 views

A dual quasi-norm (or a generalized Cauchy Schwartz inequality for quasi-norms)

Let us assume that we have a quasi-norm $\Omega(\cdot)$ on $\mathbb{R}^{p}$, now is there something like a dual quasi-norm $\Omega^{*}(\cdot)$ such that a generalized Cauchy Schwartz inequality holds: ...
5
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1answer
44 views

$E$, $F$ Banach spaces, $A: D(A) \subset E \to F$ closed densely defined unbounded operator, does $N(A) = R(A^*)$?

Let $E$ and $F$ be two Banach spaces and let $A: D(A) \subset E \to F$ be a closed densely defined unbounded operator. Does it follow that $N(A) = R(A^*)^\perp$? Notation. Let $E$ and $F$ be two ...
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40 views

can a Banach space admit a subsymmetric basis AND a symmetric basis?

Definitions. A (Schauder) basis for a Banach space is called symmetric if it is unconditional and uniformly equivalent to all its permutations. It is called subsymmetric if it is unconditional and ...
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34 views

Hermitian Projections on $C[0,1]$

A projection on a complex Banach space $X$ is said to hermitian if its numerical range is real. Does anyone know an example of an hermitian projection on $C[0,1]_{\mathbb C}=C[0,1]\oplus i C[0,1]$?
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1answer
36 views

Range of any projection is closed.

Let $X$ be a Banach space and $P$ a projection. Show that the range of any projection is a closed subspace. Can I use the fact that a Banach space is complete and thus closed and that $P = P^2$ to ...
2
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0answers
24 views

Path of completely bounded maps has uniformly bounded cb norm?

If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...