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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Why is the quotient map from a Banach space open with respect to the weak topologies?

Suppose $Y$ is a closed subspace of a Banach space $X$ and $q$ is the usual quotient map from $X$ to $X/Y$. I want to show that $q$ is an open map with respect to the weak topologies. So far I have ...
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36 views

Closed unit ball in $c_0$ is weak* dense in the closed unit ball of $l^{\infty}?$

Let $l^{\infty}(\mathbb{N})$ denotes the set of all bounded sequences, which has weak* topology as it is the dual of $l^{1}(N).$ Let $c_0$ denotes the subspace consisting of sequences which converges ...
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48 views

Cauchy-Lipschitz (or Picard-Lindelöf) theorem for Banach spaces?

Usually we meet Cauchy-Lipschitz (or Picard-Lindelöf) theorem while solving ODEs in $\mathbb R^n$. However, now I want to apply this theorem to solve a special evolutionary PDE. I look it up in ...
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1answer
28 views

Set of poinwise convergence of a sequence of weak* continuous linear functionals is weak* closed?

Let $\{\Lambda_n\}$ be a sequence of weak* cont. linear functionals on $X^{*},$ the dual of a Banach space $X.$ As every weak* cont. functinal is also norm continuous on $X^{*},$ assume that the ...
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31 views

How to see a set is clopen

Let $X$ be a manifold modeled on a Banach space. Let $E$ be a fixed Banach space, and for any point $x\in X$ there exists a local chart toplinear isomorphism $\psi_x:U_x\to\psi_x(U_x)$ where ...
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Norm of direct sum of operators acting on complemented subspaces of a Banach space.

Suppose $X$ is a Banach space with a normalized Schauder basis $\{e_k\}$ and basis constant $1$. This means for each $x$ in $X$ there is a unique sequence of scalars $\{a_n\}$ for which ...
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31 views

In the space $l_1$,$0$ is not in $conv^* \{e_i\}$

Show that in the space $l_1$ , $0$ is not in $conv^* \{e_i\}$ where $e_i$ 's are the standard basis vectors of $l_1$. I was trying to solve this by contradiction but I failed..Need some help.
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85 views

Can we classify weakly compact subsets of $l^{\infty}$?

My question is as stated. The motivation behind this is that I want to prove that such subsets are norm separable.
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92 views

Every Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$

How to prove the following mind-blowing fact? Let $(X, \|\ \|)$ be a separable Banach space, $\ell_1\subset \mathbb{R}^\infty$ - the space of absolutely summable scalar sequences. Then there ...
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1answer
70 views

In a Banach space, $x_n \to x$ and $f_n \to f$ weak* implies $f_n(x_n) \to f(x)$

Let $X$ be a Banach space. Assume that $ x_n \to x$ and $f_n \in X^*$ such that $f_n \to f$ in the $w^*$ topology on $X^*$. Show that $ f_n(x_n) \to f(x)$. I know that $f_n \to f$ in $w^*$ topology ...
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1answer
32 views

Proving that the spectral radius of a nilpotent element of a Banach algebra is equal to zero.

Let $A$ be a Banach algebra with unit $e$ and let $a\in A$ be nilpotent. I want to show that the spectral radius $\rho(a)=0$. We have that the spectral radius satisfies ...
2
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1answer
37 views

Pointwise convergence in $C(K)$ where $K$ is compact hausdorff

Let $ \{f_n\}$ be a sequence in $C(K)$ where $K$ is a compact Hausdorff space with $|f_n| \le 1$ for $n=1,2,......$ . If $f \in C(K)$ and $ f_n \to f$ pointwise on $K$ then show that there exist some ...
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44 views

Show that $T: X \to Y $ is a bounded linear operator

Let $T: X \to Y$ be a linear operator where $X$ and $Y$ are Banach spaces and $\mathcal F$ be a subset of $Y^*$ that separates the points of $Y$. Given that $f(T(x_n))\to 0$ whenever $f$ is in ...
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31 views

When is a subspace of $V^*$ dense, where $V$ is a Banach space?

Let $V$ be a Banach space. Suppose $\alpha_i$ is a collection of bounded linear functionals $V\to \mathbb{C}$ such that: $$ \bigcap_i \ker(\alpha_i) = 0. $$ Does this imply the $\alpha_i$ span a ...
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38 views

Dual space of arbitrary direct sum is the direct sum of dual spaces

Suppose that $X$ and $Y$ are Banach spaces. It is known that dual space of a finite direct sum is the finite direct sum of each dual space, that is, $$(X \oplus Y)^* = X^* \oplus Y^*$$ where $X^*$ ...
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29 views

Hahn-Banach separation theorem with a countable subset of functionals

For a separable Banach space $X$, the unit sphere of $X^*$ always contains a countable set $D$ such that $$ \left\Vert x \right\Vert = \sup_{f \in D} \left\vert f(x) \right\vert \qquad \mbox{ for ...
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What is the closure of the absolutely convex hull of this set?

Let $E$ be a Banach algebra, and $e_0\in E$, $e_{0}\neq0$. Suppose $0<r<\|e_{0}\|\leq 1$. Let $E_{1}=\{e\in E:\|e\|=1\}$. Clearly, $e_{0}\notin rE_{1}$. Consider the set ...
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Consider the Banach Space $C[0,1]$. Find decomposition of spectrum of the indefinite integral operator.

Cosider the Banach Space $C[0,1]$ of real-valued continuous function on $[0,1]$ with the supremum norm. and the linear operator $$A: x(t)\mapsto\int\limits_0^tx(s)ds$$ Find its eigenvalues, ...
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30 views

Existence of Unitary Map

I've been recently introduced to Unitary operators of a Hilbert space and I've been wondering the following. Existence of a unitary operator $T$ on a (possibly infinite) Hilbert space $H$ is simple ...
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Solution of an equation involving Banach fixed point.

I find this queistion in my textbook: QUESTION) Let $T \in \mathcal{L}(\mathcal{B})$, where $\mathcal{B}$ is a Banach Space. If exists $n$ with $||T^{n}|| < 1$, show that, given $\eta \in ...
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42 views

If $X=C[0,1]$ and $X_0=\{f\in C[0,1]|f(0)=0$, then how to show that $X/X_0$ is isometrically isomorphic to $\mathbb C$?

Of course, $X/X_0$ is the quotient Banach space with usual norm. I think it's true that the map should be one that takes a continuous function to its value at 0. But can someone give a precise ...
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56 views

$\mathcal M(K)$ is an $\mathcal{l}_1-$sum of $L_1(\mu)$ spaces

Let $K$ be a compact Hausdorff space. I want to show $\mathcal M(K)$ is an $\mathcal{l}_1$-sum of $L_1(\mu)$ spaces, where $\mathcal M(K)$ is the dual of $C(K)$. I have got the sketch of the proof but ...
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166 views

Is every Banach space densely embedded in a Hilbert space?

Can every Banach space be densely embedded in a Hilbert space? This is clear if the Banach space is actually a Hilbert space, but much can you relax this? If the embedding exists, is the target ...
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32 views

Unit ball of a subspace is weak* dense in the unit ball of its weak* closure

Let $X$ be a Banach space and $X^{*}$ its dual space. Let $M$ be a nontrivial subspace of $X^{*}$ which is not weak* closed. Let $\overline{M}^{*}$ denotes its weak* closure. I have the following ...
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31 views

Can we show that $(E\times \mathbb R)^*=E^* \times \mathbb R$ where $E$ is a Banach space?

Can we show $(E \oplus \mathbb R)^* \cong E^* \oplus \mathbb R$, where $E$ is a Banach space and $E^*$ is the dual space of $E$? What if $E$ is just a normed space or even a topological space? To be ...
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Representation of weak*-continuous linear functionals on a subspace which is not necessarily weak* closed

Let $X$ be a Banach space and let $X^{*}$ be its dual space. Let $M$ a subspace of $X^{*}$ on which a weak*-continuous linear functional $L$ is given. In the case where $M$ is weak*-closed, we know ...
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Specific Example of a bounded operator on $\ell^{p}$

Let $p\in (1,\infty)$. To construct a counter example in some other context, I am seeking a bounded sequence $(A_{n})$ of operators in $B(\ell^{p})$ with the following property. ...
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1answer
41 views

The distance from a point in $l_\infty$ to $c_0$

$l_\infty$ is the space of bounded sequences and $c_0$ is the space of sequences converge to $0$, is a closed subspace of $l_\infty$. I am trying to prove that for any $x \in l_\infty$, $\;d(x,c_0) = ...
2
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1answer
33 views

Using subspaces of $C[0,1]$ to show that Riesz's lemma is not true for $r=1$

Let $\mathcal{C}^0([0,1];\mathbb{C})$ be the set of all continuous complex-valued functions on $[0,1]$. Prove that : $a)$ $E:=\{f\in\mathcal{C}^0([0,1];\mathbb{C}) : f(0)=0\}$ is a Banach space. ...
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Is $ \left| \lim_{t \rightarrow 0}{\frac{f(x+ty) - f(x)}{t}} \right| \leq \lim_{t \rightarrow 0}{ \frac{|f(x+ty) - f(x)|}{|t|}}$

Suppose $X$ is a Banach space and $f:X \rightarrow \mathbb{R}$ is a continuous function. Is it true for all $x,y \in X$ that $$ \left| \lim_{t \rightarrow 0}{\dfrac{f(x+ty) - f(x)}{t}} \right| \leq ...
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Distance from a weak* closed subspace is achieved by an element?

Let $X$ denotes a separable Banach space and $X^{*}$ its dual space. Let $M\subseteq X^{*}$ be a weak* closed subspace of $X^{*}.$ Let $\varphi\in X^{*}\setminus M$ be given. I have the following ...
2
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1answer
81 views

Every separable Banach space is isometrically isomorphic to a subspace of $\ell_\infty$

I want to prove exactly what is written in the title, using the following idea: Let $X$ be a separable Banach space. We have that $X$ is isometrically isomorphic to a subspace $M$ of $C(B_{X^\ast})$, ...
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135 views

Does $L^p$ have a basis for which the Pythagorean identity with exponent $p$ holds?

In the $\ell^p$ spaces with $1\leq p<\infty$, let $\{e_n\}$ be the standard basis. If $x=\sum_{n=1}^\infty a_ne_n$ is in $\ell^p$, then for any $k$ we can write $$||x||^p=\sum_{n=1}^k ...
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1answer
20 views

Injectivity in Banach Space

Let X be a Banach Space, and $\phi_x:X^*\rightarrow \mathbb{C}$ be the operator defined by $\phi_x(f)=f(x)$, where $f$ is a linear bounded form in $X$. I am trying to prove that the operator ...
2
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1answer
58 views

Operator $T^*$ surjective iff $T$ topologically injective

Let $T : E \to F$ operator between Banach spaces, and $T^* : F^* \to E^*$ - adjoint operator. I want to proof next proposition: $T$ is topologically injective (or equivalently: injective with closed ...
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1answer
142 views

Is $\sup_{\| f \| \leq 1}{\left| \int f d\mu \right|} = \sup_{\| f \| \leq 1}{\{ |\mu(f)| \}}?$

The answer given by t.b. mentioned the following One of the most convenient way of writing a total variation norm is $$\| \mu \| = \sup_{\| f \| \leq 1}{\left| \int f d\mu \right|}$$ In ...
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Application of Hahn-Banach to Linear Functional Extension

I am currently enrolled in a functional analysis course and am experiencing some troubles with applying the Hahn-Banach theorem we discussed with regards to extending linear functional. In particular, ...
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61 views

Ranges of projection operators

Suppose that $X$ is a Banach space and $P$ and $Q$ be bounded linear projections on $X$ such that $PQ$ and $QP$ are compact. Does it follow that $PQ$ and $QP$ are finite-rank operators? My attempt: I ...
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$X,Y$ Banach spaces, $A\in B(X,Y)$, $\text{ran}A$ second category, then $\text{A}$ is closed. [duplicate]

this is question $III.13.1$ from Conways book on functional analysis: Suppose $X,Y$ and Banach spaces, $A\in B(X,Y)$, $\text{ran}A$ is a second category space, then $\text{ran}A$ is closed. ...
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Frobenius condition in infinite dimensions

My setting is this: $X$ a smooth manifold. $\phi_t:X\rightarrow X$ a flow of a vector field. $f\in \mathcal C^{\infty}(X;\mathbb R)$ an element of the space of smooth functions on $X$ and ...
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1answer
34 views

How to subtract infimum of a set from infimum of another set?

Suppose $A \subset M$ where $M$ is a metric space. Let $f:A \rightarrow \mathbb{R}$ be a Lipschitz map and $g:M \rightarrow \mathbb{R}$ to a map. Define $$g(y) = \inf_{x \in A}{\{ f(x) + \| f ...
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54 views

Is $(\oplus\ell_2^n)_{\ell_1}$ complemented in $\ell_1\oplus_\infty\ell_p$?

Fix any $1<p\leq 2$. Let us recall that $E:=(\oplus\ell_2^n)_{\ell_1}$ is just the space of sequences $(x_n)_{n=1}^\infty$, $x_n\in\ell_2^n$, such that $(\|x_n\|_{\ell_2^n})_{n=1}^\infty\in ...
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1answer
50 views

Compactness of the closed unit ball of ${\rm Lip}_0(X)$ in the topology of pointwise convergence implies that ${\rm Lip}_0(X)$ is a dual space

It is proven that a closed unit ball in the set of real-valued Lipschitz functions ${\rm Lip}_0(X)$ defined on a Banach space $X$ is compact for the topology of pointwise convergence. However, I ...
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1answer
33 views

Intersection of decreasing sequence of non empty closed sets

Let X is Banach space. Suppose $B_n$ are decreasing sequence of non empty closed balls. Prove their intersection is non-empty. I have some idea. Idea is pick $x_n \in B_n\backslash B_{n-1}$ in such ...
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1answer
13 views

Integration w.r.t nondegenerate Gaussian probability measure on $X$ with mean $0$

Suppose that $X$ is a Banach space. Denote $\gamma$ as a nondegenerate Gaussian probability measure on $X$ with mean $0$. Question: Is it true that $$\int_X{d\gamma(t)}=0?$$ Or we have ...
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1answer
43 views

Functor in $\mathbf{Ban}$ that puts exact sequences into exact sequences

Let $F$ - is functor in category of Banach spaces $\mathbf{Ban}$ with follow property: $f_n : A_n \to A_{n+1}$ exact sequence iff $F f_n : F A_n \to F A_{n+1}$ is exact sequence. Is it true, that $F$ ...
2
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1answer
36 views

In a Topological vector space, a subspace of codimension 1 is either dense or closed.

Let $X$ be a topological vector space and $V$ be a linear subspace of $X$ such that $\text{dim}(X/V)=1$, then either V is closed or $\overline{V}=X$. In other words if it is not closed then it ...
4
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1answer
74 views

Does there exist a Banach space with no complemented closed subspaces?

I know that every Hilbert space can be decomposed as the direct sum of two non-trivial closed subspaces, eg. taking the kernel and range of any non-trivial bounded projection. But I don't know what ...
3
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75 views

Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
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32 views

Existence of a linear map from a dense space

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...