Tagged Questions

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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Using a Quotient map to induce a well-defined map

Let $X, Y, Z$ be Banach spaces, and let $q:X\to Y$ be a quotient map. Given a bounded linear operator $T:X\to Z$, does there exist some criteria on $T$ for determining when $q$ can be used to induce ...
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$Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
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Space of $f(0)=f(1)$: Is Hilbert space?

Let $S$ be space consisting of collection of square integrable continuous functions $f:[0,1]\rightarrow\mathbb{R}$ with the constraint $f(0)=f(1)$. So $S$ is an inner product space with the inner ...
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How to prove $\limsup\limits_{n\to\infty}\rho_k(x_n+x)=\limsup\limits_{n\to\infty}\rho_k(x_n)+\rho(x)?$ on $\ell_1$

Let $p(.)$ be an equivalent norm to the usual norm on $\ell_1$ such that $$\limsup\limits_{n\to\infty} p(x_n+x)=\limsup\limits_{n\to\infty}p(x_n)+p(x)$$ for every $w^*-$null sequence $(x_n)$ and for ...
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About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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Prove or disprove: $\{t^{2k}\}_{k=0}^{\infty}$ complete in $L_2[-1,3]$

Is $\{t^{2k}\}_{k=0}^{\infty}$ not complete in $L_2[-1,3]$?(Here, completeness of a system is equivalent to the density of its span) Obviously many polynomials in the domain will be irreleant, but I ...
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When can I say that $\overline{A} \subset B$ if I know that $A \subset B$?

my question is as stated in the title: When can I say that $\overline{A} \subset B$ if $A \subset B$? Here $A,B$ are normed spaces and the closure of A is taken with respect to the norm of B. Can I ...
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Construct a sequence in Banach Space

Prove the equivalence between: $\forall x \in B_E = \{y \in E:\|y\| \leq 1 \}$ $\exists (x_n) \subset E$ such that $\|x_n\|=1$ and $x_n \rightarrow_w x$ (weak convergence). $\exists (x_n) \subset E$ ...
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Any metric space can be isometrically embed in some Banach space? [duplicate]

I have just read the question of the title in an article from Kirchheim. I didn't know this result, does any one know where I can find a proof of it?
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A Banach space $X$ is reflexive iff $X^*$ is reflexive [duplicate]

I have already shown that if $X$ is reflexive then $X^*$ is reflexive, but I need some help on the other direction. The canonical mapping is defined by $$J : X \to X^{**}, \ J(x) (f) = f(x)$$ For ...
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Projections in the lp direct sum $E=(\bigoplus_{n=1}^\infty\ell_1^n)_p$.

Fix $1<p<\infty$ and define the (reflexive) Banach space \begin{equation*}E=\left(\bigoplus_{n=1}^\infty\ell_1^n\right)_{\ell_p}.\end{equation*} Let $Y$ denote the closed subspace of $E$ ...
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Basis equivalent to the unit vector basis of $(\oplus_{n=0}^\infty\ell_\infty^{2^n})_p$

Definitions and notation. Fix $1<p<\infty$, and define the (reflexive) Banach space \begin{equation*}E=\left(\bigoplus_{n=1}^\infty\ell_\infty^{2^{n-1}}\right)_{\ell_p}.\end{equation*} It has a ...
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Is a reflexive space necessarily an L-embedded space?

I am reading some paper about L-embedded space. For the definition of L-embedded space, see http://www.sciencedirect.com/science/article/pii/S0022247X02001075. Let $Y$ be a Banach space and $P$ a ...
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Question about the proof $X'$ reflexive $\Rightarrow X$ reflexive.

I have a doubt in the proof I have been given of the fact: For a Banach space $X$, if $X'$ is reflexive then $X$ is reflexive. This is proven by showing first theorem 1 and theorem 2, which I quote ...
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Existence of a function that fulfills an equation

I am just revising for my exams and came across this question: Show that in the Banach-space of functions that are continuous in the interval $[-1,1]$, together with the supremum-norm, there is ...
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If B(X) is isomorphic to B(Y), does that mean X is isomorphic to Y (for X and Y Banach spaces)?

Let $X$ and $Y$ be Banach spaces such that $\mathcal{B}(X)$ is linearly isomorphic to $\mathcal{B}(Y)$ (where $\mathcal{B}(\cdot)$ denotes the algebra of bounded linear operators). Must it always be ...
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Second adjoint of the canonical embedding

Suppose that $X$ is a Banach space. Denote by $\kappa_X$ the canonical embedding of $X$ into $X^{**}$. Do we always have $$(\kappa_X)^{**} = \kappa_{X^{**}}?$$
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Is $AC[a,b]$ closed in $(BV[a,b],TV)$?

Consider $BV[a,b]$ the space of all bounded variation functions on a real interval $[a,b]$, endowed with the total variation norm $TV$. $AC[a,b]$, the space of absolutely continuous functions, is a ...
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Total variation on BV functions: “Banach seminorm”?

Suppose I consider the space $BV[a,b]$ of all bounded variation functions on $[a,b]$ a real interval. I endow it with $\|f\|=TV(f)$ the total variation norm. Do I get a Banach space? How can I prove ...
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Summation functional on a Hamel basis

Let $X$ be an infinite-dimensional Banach space. Is it possible to choose a Hamel basis $B$ of $X$ such that the linear functional defined by $f(b)=1$ ($b\in B$) was continuous?
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Showing that $\log (\log 1+\frac{1}{|x|})$ belongs to $W^{1,p}(\Omega)$ for $p \geq 2$.

I want to show that the function $f$ belongs to $W^{1,p}(\mathbb{R}^n)$ for $p \geq 2$, where $f$ is defined as $$f(x)=\log\left(\log \left(1+\frac{1}{|x|}\right) \right)$$ Note: This is an example ...
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Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
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Relationship between spectral rays commuting

Show sing binomial Newton that if $S$ and $T$ are continuous operators on Banach space $B$ such that $ST = TS$ then $$r_\sigma(T + S) \leq r_\sigma(T) + r_\sigma(S)$$ where $r_\sigma$ is spectral ...
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$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?
Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ? I know that it is true if $B$ is a Hilbert space ( we only need an inner product ...
In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?
Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance