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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4
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169 views

Non strictly singular operators

Let $X$ be a separable Banach space and let $T:X\to X$ be a bounded operator that is not strictly singular. Can we always find an infinite dimensional, closed, and complemented subspace $Y$ of $X$ ...
0
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2answers
124 views

Neighborhood base of weak topology

If $X$ is a Banach space, the weak topology on $X$ is the weakest topology in which each functional $f$ in $X^\ast$ is continuous. I have some difficulties in understanding its neighborhood basis in ...
12
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1answer
605 views

Are the coordinate functions of a Hamel basis for an infinite dimensional Banach space discontinuous?

The question is in the title really, but I suppose I could at least fix some notation here. Let $X$ be an infinite dimensional Banach space - over the reals for the sake of concreteness. Use choice ...
1
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2answers
29 views

Does for $T \in B(X)$ with $\|T\|>1$ exist $T^{-1}$?

Is it true if $\|T\|>1$, where $T \in B(X)$ for some Banach space $X$, then $T^{-1}$ exists? I suppose that for $\|T\|=1$ this isn't true? Because, if we suppose that inverse exists for such ...
0
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0answers
17 views

Showing the space of $\beta$-operators between Banach lattices is a Banach space [on hold]

Let $E,F$ be a Banach lattices. A linear map $T:E \rightarrow F$ is called $\beta$-operator if $\lVert T\rVert_\beta < \infty$, where $$\lVert T\rVert_\beta := \sup \bigl\{ \bigl\Vert ...
2
votes
1answer
223 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 ...
1
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1answer
25 views

Please give me an example of closed subspace of banach space under some conditions

Please give me one example of Banach space $X$ and its closed subspaces $S,T,U$ which suffice following conditions. Any of $S+T,T+U,U+S$ is not a closed subspace of $X$. I can say there are some ...
0
votes
0answers
41 views

Convergence of sequence as $O(1/n)$ using damped iterations

I am trying to understand the argument for convergence here (page 16) for damped iterations of non-expansive maps. Say we have a sequence $\{x^n\}_{n=0}^{\infty}$ that is generated as $x^n = \theta ...
6
votes
1answer
43 views

Finding a norm making a subspace dense

Suppose $V$ is a (real or complex) vector space and $W$ is a subspace of $V$. Under what conditions is there a norm on $V$ making $W$ a dense subspace of $V$? That $V$ and $W$ have the same ...
2
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2answers
41 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwarz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
3
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0answers
59 views

When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
0
votes
2answers
32 views

Fixed-point analysis similar to Banach Fixed Point Thm

I have a fixed-point question similar to the Banach fixed-point theorem. Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known ...
1
vote
1answer
26 views

Continuity of multiplication of operators in the strong operator topology - find an error

I need help in finding the mistake in the following reasoning. I proved that if dimension of Banach space $X$ is infinite, then multiplication of bounded operators is separately continuous but not ...
0
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1answer
22 views

Uniform Boundedness: Nets

I thought so far that the uniform boundedness principle applies (according to the proof I know) to any net of bounded operators. Now I read that this works for sequences only. Can you shortly explain ...
11
votes
2answers
3k views

Compact operator maps weakly convergent sequences into strongly convergent sequences

I found the following property of compact operators in a proof, and I can't prove it. Prove that if $T \in \mathcal{L}(E,F)$ is compact, and if $u_n \rightharpoonup u$ (the sequence converges ...
0
votes
1answer
30 views

Summability: Equivalence

Summability Given a Banach space $E$. Consider sums: ...
6
votes
1answer
58 views

Do we need completeness for a weak*-convergent sequence to be bounded?

Let $(\phi_n)_n$ be a weak* convergent sequence in the dual of some normed space $X$ with (weak*-)limit $\phi$. If $X$ is Banach then it follows from the uniform boundedness principle that $\sup_n ...
1
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0answers
25 views

Norm of a product of projections in a Banach space

Let $X$ be a Banach space and let $P_1,P_2$ be two projections in $B(X)$, i.e., $P_1^2 = P_1, P_2^2=P_2$. My question: under what conditions do we have that $\Vert P_1 P_2 \Vert = \sqrt{\Vert P_2 ...
3
votes
1answer
68 views

The norm of linear functional $x\mapsto \sum_{n=1}^{\infty} \frac{x_n}{2^n}$ on $c_0$

Consider the mapping $\phi :c_0 \to \mathbb{R}$ defined by $\sum_{n=1}^{\infty} \frac{x_n}{2^n}$. Compute $\|\phi\|$ Does there exist a $x \in c_0$ such that $\|x\|=1$ and $\|\phi\|=|\phi(x)|$ ...
2
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0answers
28 views

Equivalent definition of uniform convexity

A Banach space $X$ is said to be uniformly convex if the following is satisfied: For $\epsilon>0, \exists \delta>0$ such that $x,y\in X, \|x\|, \|y\|\leq 1$, $\|x-y\|\geq \epsilon \Rightarrow ...
3
votes
1answer
66 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
3
votes
1answer
24 views

Bessel potential space: Proof of completeness

I want to know a proof that the (one-dimensional) Bessel potential space (for $p=2$) $$H^s(\mathbb{R})=\left\{f:\mathbb{R}\to\mathbb{C}:\int_{\mathbb{R}}(1+\lvert \xi\rvert^2)^{\frac{s}{2}}\lvert ...
2
votes
2answers
77 views

Proving that a space is complete

There is something that bugs me about the proof I've been shown that $C(\Omega)$ (the space of continuos function on $\Omega$, a compact subset of $\mathbb R^n$) with the $\sup$ norm is complete. ...
4
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0answers
30 views

A question about equivalence of norms involving infimum

Let $I$ be a Banach space with norm $\lVert\cdot\rVert_I$. The norm $$\inf\{\lVert(G_i(u_i))_i\rVert_{\ell^2}\mid u=\sum_{I \geq 0}u_i\}\qquad\text{is equivalent to}\qquad \lVert{u}\rVert_{I}$$ where ...
-1
votes
2answers
33 views

Difference quotients and weak derivatives (Evans 5.8.2 theorem 3) [closed]

Could anyone give a proof on the remark below the theorem? Basically it is problem 11 I think the proof relates to example 19.19 (p.375) in enter link description here But I really do not ...
3
votes
1answer
54 views

Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
6
votes
1answer
103 views

Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$

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

Borel Sets and Translations

Suppose $\mathcal{F}$ is Borel $\sigma$ algebra for a separable Banach space $X$ (i.e., potentially infinite dimensional). Is it obvious that for any $A \in \mathcal{F}$ and any $a\in X$, the ...
0
votes
0answers
49 views

Prove exponential $e^f$ is of class $C^\infty$

Let $E$ a Banach space, $F=L(E,E)$ of linear and continuous functions. Define $f^0={\rm id}_E$, $f^n=f\circ\cdots\circ f$, $n$ times. Put $\exp(f)=\sum_{n=0}^\infty \frac{f^n}{n!}$. How to show the ...
1
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1answer
38 views

What is the importance that an assumption needs to state whether a space is Banach space?

I am self studying functional analysis and I don't not see the utility of authors trying make it clear that a space $X$ is a Banach space before proceeding with a definition. For example, going ...
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2answers
26 views

Why does it mean for Banach space to be a locally convex topological vector spaces

A Banach space is simply a complete normed linear space. According to Wikipedia it is also a locally convex topological vector space. How does complete + normed + linear space translate into locally ...
1
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1answer
41 views

Existence of unbounded operators on Banach spaces

I'm confused by the questions Discontinuous linear functional and Example of an unbounded operator which ask about unbounded linear functionals/operators on Banach spaces. I don't understand how ...
1
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1answer
19 views

Passing complemented subspaces to duals

Sorry for this rather basic question from Banach space theory. Suppose I have a complemented subspace $E$ of a Banach space $X$. So let's write $i:E\to X$ to be the inclusion map. Then I have a ...
0
votes
1answer
24 views

Composition of $C^\infty$ maps between Banach spaces is $C^\infty$.

Let $V$, $W$, and $X$ be Banach spaces, and let $A \subset V$ and $B \subset W$ be open. Suppose that $F \in C^\infty(A,W)$, $G \in C^\infty(B,X)$, and $F(A) \subset B$. Is there a non-combinatorial ...
1
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1answer
26 views

cauchy's integral formula in operator theory

Can you prove cauchy's integral formula based on the assumptions in Conway book, operator theory, VII, 4.2? 4.2. Cauchy's Integral Formula If $\mathcal X$ is a Banach space, $G$ is an open subset ...
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votes
1answer
21 views

Operator not bounded below on sum of subspaces

Let $X$ be a Banach space. Say that a bounded linear operator $T\colon X\to X$ is bounded below by $\delta>0$ on $Y\subset X$ if $\|Tx\|\geqslant \delta \|x\|$ for all $x\in Y$. Is there a Banach ...
1
vote
1answer
22 views

Embedding: Extension

Problem Given Banach spaces $E_0$ and $E$ Regard dense domain: $$\overline{\mathcal{D}_0}=E_0\quad\overline{D}=E$$ Consider an embedding: ...
1
vote
1answer
185 views

Is this proof correct? (left inverse and topologically complementary subsets)

I want to prove the following theorem: Theorem. Assume $T \in \mathcal L ( X, Y )$ is injective. The following statements are equivalent: $T$ admits a left inverse; Im($T$) is closed and ...
5
votes
1answer
118 views

Every closed subspace of ${\scr C}^0[a,b]$ of continuously differentiable funcions must have finite dimension.

If $F \subset {\scr C}^1[a,b] \subset {\scr C}^0[a,b]$, then $\dim F < +\infty,$ where $F$ is a closed subspace (in $ {\scr C}^0[a,b]$). I found this answer, which is very good and solves the ...
1
vote
1answer
36 views

How $\|a_x\| \leq c$ became $\|a_x\| \leq c\|x\|_E$?

Reading this answer here, I didn't understand the last $\color{red}{\leq}$ in: $$\Vert a(x,y)\Vert_G=\Vert a_x(y)\Vert_G\leq\Vert a_x\Vert\Vert y\Vert_F\color{red}{\leq} c\Vert x\Vert_E\Vert ...
16
votes
1answer
279 views

Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?

It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
2
votes
2answers
58 views

Banach spaces containing copies of $\ell^1$

Why is it important that a Banach spaces $X$ contains (or not) copies of the space $\ell^1$? I always hear talk about it but I don't know its importance. Could someone explain this?
0
votes
1answer
20 views

$p$-operator space property

If $S,T,U,V\in B(L_p(X,\mu))$, $p\in[1,\infty)$, and we regard $\begin{pmatrix} S & T \\ U & V \end{pmatrix}$ as an operator on $B(L_p\oplus_p L_p)$, then supposedly we have ...
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votes
1answer
54 views

Seminorms: Dual Space

Remarks This thread is only Q&A!* *(See guidelines: Q&A) Problem Given a Banach space $E$. Consider its dual space: $$E':=\mathcal{C}(E,\mathbb{C})\cap\mathcal{L}(E,\mathbb{C})$$ Regard ...
0
votes
1answer
35 views

Fraction of Lipschitz functions among absolutely continuous ones

Is it true that the space of Lipschitz functions on $S^1$ is a $G_\delta$ subset of the space of absolutely continuous functions on $S^1$? In which topologies ($L^p$, uniform, $C^k$, etc) it is true? ...
0
votes
1answer
36 views

Is the set defined by inequality $\|Tx\|^2\leq\|T^2x\|\|x\|$ a subspace of a Banach space?

Let $X$ be a complex Banach space and $T$ be a bounded linear operator on $X$. Put $Y=\{x\in X:\|Tx\|^2\leq\|T^2x\|\|x\|\}$. Is $Y$ a subspace of $X$? I know is that $Y$ is closed and $aY$ is ...
1
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0answers
20 views

A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
2
votes
1answer
59 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
0
votes
1answer
15 views

Codimension of the image of the polynomials subspace is infinite

Consider the interval $I=[0,1]$ and the Banach space $E$ of real continuous functions defined on $I$ ($E=\mathcal C_{\mathbb R}(I))$. $P \subset E$ is the subspace of polynomial functions (restricted ...
5
votes
3answers
380 views

Is $\ell^1$ isomorphic to $L^1[0,1]$?

Can there be a continuous linear map, with a continuous inverse, from $l^{1}$ to $L^{1}(m)$ where $m$ is the Lebesgue measure on the unit interval $\left[0,1\right]?$ My thinking to this should be ...