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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Sum of bounded and unbounded operators

Is there a Banach space $X$, $S$ an unbounded operator defined on a dense subspace $D$ of $X$ and a bounded operator $T$ on $X$ such that $$S+T|_D$$ is bounded? What if $T$ is assumed to be ...
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325 views

An application of J.-L. Lion's Lemma

Let $X,Y$ and $Z$ be three Banach spaces with norms $\|\cdot\|_X$, $\|\cdot\|_Y$,$\|\cdot\|_Z$. Assume that $X\subset Y$ with compact "injection" and that $Y\subset Z$ with continuous injection. Then ...
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275 views

Banach spaces isomorphic to square

This is another exercise from Allan's book "Introduction to Banach Spaces and Algebras". Exercise 2.9: A Banach space $E$ is said to be homeomorphic to its square if $E\oplus E$ is linearly ...
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215 views

$\ell_p$ sums of Banach spaces

Let $p\in (1,\infty)$ and let $(E_\alpha)_{\alpha<\omega_1}$ be a family of Banach spaces. Set $E=\left(\bigoplus_{\alpha<\omega_1}E_\alpha\right)_{\ell_p(\omega_1)}$. Must $E$ be isomorphic to ...
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344 views

Is kernel a complemented subspace

Let $\mathcal{A}:X\to Y$ be continuous linear operator, $X$ and $Y$ are Banach spaces. Let $\text{Im} \mathcal{A}=Y$. Is $\ker\mathcal{A}$ a complemented subspace of $X$?
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Bochner integral = 0 iff $f = 0$

This problem is about integrals of functions taking values in a Banach space. Let $f \in L^1(X,S,\mu,B)$ where $X$ is a set with a $\sigma$-algebra $S$ and a measure $\mu$. Function $f$ takes ...
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208 views

Principle of Local Reflexivity

I'm having a hard time trying to understand a proof of the Principle of Local Reflexivity. I'm following the proofs from 1) Topics in Banach Space Theory (by Fernando Albiac, Nigel J. Kalton) 2) ...
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306 views

How to show that the sum of $L^p$ spaces is Banach.

Let $p<q$ be positive integers (with the allowance that $q$ may be $\infty$). How can we show that the sum of $L^p$ and $L^q$ is a Banach space under the norm $\|f\|=\inf\{\|g\|_p+\|h\|_q: ...
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1k views

Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?

(ZFC) Let $ \big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $ be a Banach space. Define $ \mathbf{B} \; = \;\big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $. Define $\: \mathbf{B}_0 ...
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839 views

Isometric Immersion of a separable Banach Space into $\ell^{\infty}$

The problem is: Let $X$ be a separable Banach space then there is an isometric immersion from $X$ to $\ell^{\infty}$. My efforts: I showed that there is an isometry from $X^*$ (topological dual) to ...
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1answer
281 views

Continuous functions and restrictions to a countable domain

Let $I$ a compact set and $f\in\mathcal{C}([0,1]^I)$. Then exists $J\subset I$, countable or finite, such that: if $x,y\in [0,1]^I$ such that $x\big|_J=y\big|_J\Rightarrow f(x)=f(y)$.
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Is there a null sequence that is in no $\ell_p$ with $p<\infty$?

Is $\bigcup_{p<\infty}\ell_p=c_0$? At least one inclusion obvious: every p-summable sequence converges to zero.
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Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$? In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ ...
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397 views

Complemented subspace of a Banach space

I have a quick question, If $E$ is a Banach space and $H$ is a closed subspace of $E$, could we affirm this proposition: If exists a linear continuous function $S:E\to H$ such that $S\circ i ...
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65 views

Is the set $E$ of sequences containing only entries $0$ and $1$ in $(m,\left \| \cdot \right \|_\infty)$ complete?

I can't really wrap my head around $E$, or a Cauchy sequence in $E$. I need to take a Cauchy sequence in $E$ and show it's Cauchy in $(m,\left \| \cdot \right \|_\infty)$? I think I can show $(m,\left ...
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Example of a closed subspace of a Banach space which is not complemented?

In this post, all vector spaces are assumed to be real or complex. Let $(X, ||\cdot||)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called $\underline{\mathrm{complemented}}$, if there ...
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96 views

Sufficient conditions for an operator to have complemented image

Given a bounded operator $A\colon X\to Y$ ($X$, $Y$ - Banach spaces) with $A^*\colon Y^*\to X^*$ being an isomorphism onto its range. Under which assumptions on $A:X\to Y$, the range of $A$ is ...
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123 views

Classifying a point on the unit ball of a normed space as extreme depending on the field

Sorry for the awkward title. I'm looking for an example of a normed space $X$ and a point $x$ on $X$'s unit ball such that $x$ is an extreme point of the unit ball when considering $X$ as a vector ...
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64 views

Disjointly supported functions

Let $(f_n)_n$ be sequence of real-valued continuous functions on a compact, Hausdorff space $K$ with pairwise disjoint (closed) supports satisfying $$0<\inf_n \|f_n\|\leq \sup_n\|f_n\|<\infty.$$ ...
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296 views

Simple example of not a Banach space. Product topology.

Claim: $$R^\infty \text{ is not a Banach space when equipped with its natural product topology}$$ I need help proving this 'obvious' claim. I just got acquainted with a definition of a product ...
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1answer
285 views

A question about the coercivity of a lsc and convex function.

I was doing a proof and I need to show a result to conclude it: $X$ is a reflexive Banach space with a norm, $\|\cdot\|$, of class $\mathcal{C}^1$. $f:X\to\overline{\mathbb{R}}$ is lower ...
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1answer
206 views

Second adjoint operators

Let $X$ be a non-reflexive Banach space. Is it possible to decompose the identity operator $I_X=T+S$ where $T,S$ are bounded operators on $X$ with the property that $T^{**}x^{**}, S^{**}y^{**}\in X$ ...
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718 views

How do you prove that $\ell_p$ is not isomorphic to $\ell_q$?

I guess that for all $1\le p,q<\infty $, such that $p\ne q$ , the spaces $\ell_p$ and $\ell_q$ are not isomorphic, but how do you prove this?
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469 views

Differentiability of Moreau-Yosida approximation.

I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...
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1answer
206 views

Monotone convergence to a fixpoint in a Banach space

Let $\mathscr X$ be a complete separable metric space and $\mathbb B$ be the Banach space of all real-valued bounded measurable functions on $\mathscr X$. The partial order on this space is introduced ...
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1k views

The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
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1answer
320 views

Looking for a proof of the completeness of $C^{n}[0,1]$.

Can someone refer me to a verification of the completeness of $C^{n}[0,1]$ under the norm $\|f\|_{C^{n}} = \sum_{k=0}^{n}\|f^{(k)}\|_{\infty}$? I tried to follow the same approach as the standard ...
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1answer
129 views

Bounding the norm of an extension of a linear function from a subspace of a normed space to a finite dimensional normed space

I got this problem in my homework. Let $X$ be a normed space, and let $X_0\subset X$ be a subspace. Let $T$ be a continuous linear function from $X_0$ to $Y$, where $Y$ is an $n$-dimensional normed ...
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80 views

Using convexity and separation to prove bounds on norm bound functionals

I'm quoting here a homework problem with two clauses. I've already managed to find a solution for the first clause, and have problems generalizing it for the second clause, I'll go into details after ...
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851 views

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
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1answer
171 views

If $T$ has finite rank, then: $I-T$ is injective if and only if $I-T$ is surjective?

I have a Banach Space $X$ and an linear continuous operator $T\colon X\to X$ that has finite rank (i.e. $\dim {T(X)}<\infty$). Then, $I-T$ is injective if and only if $I-T$ is surjective?
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Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
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758 views

Where does the theory of Banach space-valued holomorphic functions differ from the classical treatment?

For a Banach space $V$ over $\mathbb{C}$ and $U \subset \mathbb{C}$ open, one can easily check that the notions of holomorphy hold for maps $f: U \rightarrow V$ just as in the classical sense. Indeed, ...
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321 views

Reflexive Banach algebras?

I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces. For a Banach space $X$, we investigate its dual $X'$ ...
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491 views

Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor ...
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1answer
565 views

Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However it seems Folland does not give many examples to illustrate the motivation behind much of ...
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375 views

Banach-Saks property and reflexivity

On the German Wikipedia page on the Banach-Saks property, they claim that every Banach space with the Banach-Saks property is reflexive but that the converse is not true. There should be a proof due ...
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1answer
285 views

a question about invertibility of Banach Algebra

If $X$ is a Banach algebra with identity, and $0$ is the only element $x \in X$ such that there is a sequence $\{ {x_n}\} \subset X$, $\left\| {{x_n}} \right\| = 1$ and $x{x_n} \to 0$ or ${x_n}x \to ...
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1answer
104 views

Second conjugate operators and their representations

We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,n\geq 1}$ which acts on a sequence $a=[a_1, a_2, a_3, \ldots ]\in c_0$ in the same way as usual ...
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1answer
514 views

Swapping a limit and a $\sup$

In this proof of the completeness of $(C(K), \| \cdot \|_\infty)$ they use the following inequality: $$ \sup_{x \in K} \lim_{m \to \infty} | f_n(x) - f_m(x) | \leq \liminf_{m \to \infty}\, \sup_{x ...
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244 views

To construct a counterexample of normed space

Please construct a counterexample for the following: $A$ is normed space and $M$ is a dense subspace of $A$, if there is a functional $f$ such that $f(M) = 0$, then $f=0$. Besides, if $A$ is a Banach ...
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How to prove this metric space is a complete metric space?

How to prove the metric space $L^{p}[a,b]$ is a complete metric space using the definition that says, Every Cauchy sequence in the metric space should converge to some point in that space? ...
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82 views

Biduals of operators

Let $T\colon X\to Y$ be a bounded linear operator acting between Banach spaces. Suppose $T$ is an isomorphism onto its range. Must $T^{**}\colon X^{**}\to Y^{**}$ be an isomorphism onto its range ...
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259 views

Operator norm. Alternative definition

Let $T\colon X\to Y$ be a linear operator with norm $$\|T\|=\sup_{\|x\|=1}\|Tx\|.$$ Prove that $$\|T\|=\sup_{\|x\|\leq 1}\|Tx\|.$$
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How to prove this inequality in Banach space?

In a normed space $(E,\lVert \cdot\rVert)$ space we have the following inequality: $$\forall\, x,y\in E,\quad\|x\|^{2}-\|y\|^{2}\leq \lVert x-y\rVert\cdot \|x+y\|.$$ How can we prove it?
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268 views

Why is $GL(B)$ a Banach Lie Group?

Banach Lie Groups are what you'd expect: http://www.encyclopediaofmath.org/index.php/Lie_group,_Banach If $B$ is a Banach algebra then why is $GL(B)$, the set of invertible elements of $B$, a ...
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117 views

a condition for $X\neq X^{**}$ when $X$ is a Banach space

Let $\left(X,\|\cdot\|\right)$ be a Banach space. I need to show that if $\exists f:X \to K$ ($K$ is either the real or complex numbers) a bounded linear functional s.t $\forall x\in X \setminus \{ ...
12
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828 views

Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
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An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ Setup: Let $l^\infty$ be the set of bounded sequences (with terms in $\mathbb{R}$), and let $l^1$ be the set of sequences ...
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72 views

Image of a Markushevic basis

Let $X, Y$ be Banach spaces and let $T\colon X\to Y$ be a bounded linear operator. Assume that $X$ admits a Markushevic basis. Does $\overline{T(X)}\subseteq Y$ admit a Markushevic basis as well? ...