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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Verification: Closed Set Expands to Fill Space, but Contains No Open Ball $B_\epsilon(0) $?

I have the proof that $C$ closed, convex, symmetric in Banach space $X$ and $\cup_{n \in N \setminus 0} n.C= X $ then $B_\epsilon(0) \subset C$ for some $\epsilon > 0$. I also have the proof for $...
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A reflexive subspace of $\ell_p(c_0)$

Fix $1<p<\infty$ and define the Banach space \begin{equation*}\ell_p(c_0)=\left\{(x_n)_{n=1}^\infty\subseteq c_0:\left(\|x_n\|_{c_0}\right)_{n=1}^\infty\in\ell_p\right\}\end{equation*} endowed ...
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$C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Consider $C([0, 1])$, the linear space of continuous complex-valued functions on the interval $[0, 1]$, with the norm $$\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx.$$ I have to ...
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About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
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on $a$ proof that every normed linear space admits a completion.

One of the proofs that any metric space $M$ has a completion is to use the fact that the space $C_\infty (M)$ of all complex-valued continuous and bounded functions on $M$ is complete with respect to ...
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Proof Verification: C closed, convex, symmetric in Banach space X and $\cup_{n \in N \setminus 0} n.C= X$ then $B_\epsilon(0) \in C $.

I have an outline of the proof of this which I've expanded (correctly or otherwise) below, I'd appreciate feedback on it. (I think that C has to be closed in order to assert that $\cup_{n \in N \...
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1answer
24 views

Show space of C1 functions on (0,1) is a Banach lattice

I'm working on the beginning of a book on Banach lattices, and it wants me to show that $C^1(0,1)$ is a Banach Lattice, with the norm $\|f\|=\|f'\|_\infty+|f(0)|$ and the order $f\le g$ if $f(0)\le f(...
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a proof that $c_0$ is a Banach space

I'm currently reading the functional analysis lecture notes taught in MIT, and I came across a filling-in on the part of the reader. I wonder if I've filled in the details as expected by the author. I ...
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If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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1answer
46 views

Is $C[a,b]$ isomorphic to $C([a,b]\cup[c,d])$

It is very simple to prove that $C[a,b]$ is isomorphic to subspace of $C([a,b]\cup[c,d])$ and vice versa $C([a,b]\cup[c,d])$ is isomorphic to subspace of $C[a,b]$. Is it true for $\bigl(C[a,b], C([a,...
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1answer
49 views

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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2answers
34 views

$Im(K)\subset Y$ 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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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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143 views

Proving that $c$ is a Banach space.

I want to prove that $$c = \{ (x_n)_{n\geq 0} \mid x_n \in \Bbb C \text{ and the sequence converges} \}$$ with the norm $$ \left\|(x_n)_{n\geq 0}\right\|_{\infty} =\sup_{n\geq 0} |x_n|$$ is a Banach ...
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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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The space of functions that vanish at infinity is a Banach space

Prove that $C_0(X) = \{ f \in C(X) \mid \forall \varepsilon >0 \ \exists K \subset X \text{ compact such that } |f(x)| < \varepsilon \text{ for } x \notin K \}$ is a Banach space with the norm $\...
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2answers
52 views

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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1answer
40 views

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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1answer
37 views

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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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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1answer
184 views

Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set

I'm trying to prove the following: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set. I came up with the following idea: Let $ (X,d) $ be a ...
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1answer
18 views

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 is reflexive if and only if its dual is reflexive

How to show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive?
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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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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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1answer
119 views

Reflexive subspaces of $\ell_{\infty}$

Let $\ell_{1}$ and $\ell_{\infty}$ be the usual spaces of, respectively, convergent and bounded sequences, with their usual norms. As we know, the topological dual of $\ell_{1}$ is $\ell_{\infty}$, ...
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21 views

Image of unit ball precompact implies bounded

I need to prove that if the image of the unit ball $X_1$ of a Banach space $X$ along an operator $A:X\rightarrow Y$ is precompact, $A$ is bounded. Is my solution correct? $\|A\|=\sup_{\|x\|=1}\|Ax\|...
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1answer
69 views

Determining whether equality $ \|T v\| = \|T\| \cdot \|v\|$ is possible

As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible ...
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Implicit function theorem for Banach spaces

I was wondering if someone could give a bit of broad advice regarding working with Implicit Function Theorem (IFT) and, I guess, the Catastrophe theory. This is something completely new to me. ...
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29 views

What's the difference between the topology defined by a seminorm and the topology defined by the norm it induces?

I was just wondering whether there's some big difference between the topology generated by a seminorm and the norm it induces. For instance, Suppose $X$ is normed and $A$ is a subspace. $X/A$ is semi-...
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1answer
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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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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 $\|e_1+e_2+\cdots+e_n\|\leq C$ for all $n$ then $(e_i)_{i=1}^n$ is uniformly equivalent to the basis of $\ell_\infty^n$?

Conjecture. Let $n\in\mathbb{N}$, and let $(e_i)_{i=1}^n$ be a normalized unconditional basis for an $n$-dimensional Banach space $E_n$ with Schauder basis constant $K\in[1,\infty)$. If $\|e_1+e_2+\...
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1answer
25 views

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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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 ...
2
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1answer
38 views

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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Showing basic properties of Riemann integral

I'm (recreationally) trying to see to what extent Riemann integration can be extended to functions of the form $f : [a, b] \to B$ where $[a, b] \subseteq \mathbb{R}$ is some compact interval and $(B, \...
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56 views

A corollary of the Hahn-Banach theorem

Let $Z$ be a subspace of normed linear space $X$ and that $y$ is an element of $X$ whose distance from $Z$ is $d$. Then there exists a $\Lambda \in X^* $ (the dual space of $X$) so that $\| \Lambda\| \...
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Weak convergence $\iff$ strong convergence in finite dimensional space

I am seeking a proof of the following claim. Weak convergence $\implies$ strong convergence in a finite-dimensional normed linear space. Thank you.
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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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1answer
53 views

Banach-Space-Valued Analytic Functions

This is Chapter VII, $\S$3, exercise 4, from Conway's book: A Course in Functional Analysis: Let $X$ be a Banach space and $G\subset \mathbb{C}$ an open subset. We say that $f: G \to X$ is analytic ...
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How to use the completeness here

Suppose that $G\subset \mathbb{C}$ is an open subset and that $X$ is a Banach space. Fix $z_0 \in G$ and let $V:=\{(h,k) \in \mathbb{C}^*\times\mathbb{C}^* : z_0+h \in G\ ; z_0 + k \in G\}$. Let $f: G ...
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Do null sequences in Banach space have summable subsequences?

One of the very nice features of null scalar sequences is the fact that they admit summable subsequences. Is the same true in Banach spaces? That is, if $(x_n)_{n=1}^\infty$ is a sequence in a Banach ...
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Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
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Fixed points and banach spaces [closed]

Let $B$ be a closed ball centered at $0$ in a Banach space $E$ and $F:B\to E$ be a contractive map such that $F(x)=-F(-x)$ for every $x\in\partial B$. Show that $F$ has a fixed point. Any ideas?
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Uncountable sets in metric spaces

Suppose we work in a separable, complete metric space $X$. Let $Z$ be an uncountable subset of $X$, must there exist $x_0\in Z$ and a sequence $(x_n)_{n=1}^\infty$ of elements in $Z$ different from $...
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1answer
34 views

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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21 views

Strongly measurable implies Borel measurable in separable space

Let $(M,\mu)$ be a measure space, $X$ be a Banach space, $f: M \to X$ be a function. $f$ is said to be strongly measurable if there is a sequence of simple functions $\{f_n\}\to f$ pointwisely a.e.. $...
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About a partial converse to the Banach-Steinhaus Theorem

I've been reading the GTM text Topics in Banach Space Theory by Albaic and Kalton. In the appendix, it states the following partial converse to the Banach-Steinhaus theorem: Let $\{S_n\}$ be a ...