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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Tensoring subspaces

Let $X$ be a Banach space, $E\subset X$, be a subspace and let $\hat{\otimes}$ denote the projective tensor product. Denote $L_1 = L_1 [0,1]$. Does $E\hat{\otimes} L_1$ embed into $X \hat{\otimes} ...
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705 views

Compact subspace of a Banach space .

The following statement doesn't make sense to me, can someone justify it to me ? If $K$ is a compact subset of a Banach space $Y$ then there exists for $\epsilon > 0 $ a finite dimensional ...
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207 views

Functional analysis-Closed graph therem

Let $ X$, $ Y$, $ Z$, be Banach spaces and let $ T:X\to Y $ and $ S:Y\to Z $ be linear transformations.Suppose that $S$ is Bounded and injective and that $ S \circ T $ is bounded.Prove that $T $ is ...
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619 views

Let $X$ and $Y$ be Banach spaces, show that if they are isomorphic, then $X$ is reflexive iff $Y$ is reflexive.

I want to show that if $X$ and $Y$ are two Banach spaces, and $T : X \to Y$ is an isomorphism, then $$ X \textrm{ reflexive} \iff Y \textrm{ reflexive}. $$ I saw several proofs, but I cannot ...
4
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1answer
189 views

Absolute norms and 1-unconditional sums

Absolute norm Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that $$ \|(x,y)\|_N=N((\|x\|, ...
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576 views

Is the inclusion $C^1[0,1]\subset C[0,1]$ compact?

I am working on this problem but i couldn't succeed . Consider the space $C^1[0,1]$ with the norm $$\|f\|=\max \{\|f\|_{C[0,1]}, \|f'\|_{C[0,1]}\},$$ I don't know if the inclusion map is compact, ...
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1answer
1k views

When to use Closed Graph Theorem vs. Uniform Boundedness Theorem?

I run in to problem that I often know is solvable with either the Closed Graph Theorem or Uniform Boundedness Theorem. I seem to mix up the solutions. Are there any hints on when to use which? Or can ...
10
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1answer
202 views

Growth $\beta X\setminus X$ of a Banach space $X$

Is there an analytic characterisation of the Čech-Stone compactification (in the norm topology, which is a normal space) of a Banach space $X$? The reason I ask is because I want to know what the ...
4
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83 views

The control of norm in quotient algebra

Let $B_1,B_2$ be two Banach spaces and $L(B_i,B_j),K(B_i,B_j)(i,j=1,2)$ spaces of bounded and compact linear operator between them respectively. If $T \in L(B_1,B_1)$, we have a $S \in K(B_1,B_2)$ and ...
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Weak limit and strong limit

Let $X$ be a Banach space and let $x_n \overbrace{\rightarrow}^w x$ and $x_n \overbrace{\rightarrow}^s z$ can we then say that $x = z$? My try: $$\| x- z\| = \sup_{\ell \leq 1} |\ell(x-z)| = ...
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1answer
350 views

Linear functional and convergent series in $\ell^\infty$

Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm and let $c,c_0$ be the subspaces of sequences that are convergent, resp. convergent to zero. Show that: The linear ...
3
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1answer
293 views

Submersion Theorem for Banach Spaces

I'm having difficulty proving a well-known result from functional analysis. Any hints would be greatly appreciated. Fix a Fréchet differentiable map of Banach spaces $g: X \to B$. Assume that, at a ...
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1answer
69 views

Uniqueness of for integration functional

Let $f\in C([0,1])$ and assume that there exists a positive constant C such that $\left| \int_0^1p'(t)f(t) dt \right| \leq C\|p\|_2 $ for all polynomials $p$, where $\|p\|^2_2 = \int_0^1 |p(t)|^2 dt$. ...
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1answer
153 views

Continuous maps in from Banach space to $\ell ^\infty$

Let $X$ be a Banach space. Prove that a linear map $M\colon X\mapsto \ell^p, \; p\geqslant 1$ is continuous iff for every sequence $(x_k)$ that converges in $X$ to $x \in X$, we have that the $n$-th ...
2
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1answer
110 views

Continuous mapping between $X$ and $C([0,1])$

Let X be a Banach space.Prove that a linear map $M:X \rightarrow C([0,1])$ is continuous iff for every $t\in [0,1]$, the rule $x \mapsto (Mx)(t)$ definies a continuous linear functional on X. My try: ...
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4answers
87 views

$\ell x = 0$ for all $\ell \in X'$ for banach spaces

I guess this is probably asked before but I can not find it. Let $X$ be a Banach space, and let $\ell x = 0$ for all $\ell \in X'$. Then $x = 0$ If all projections $\pi_\alpha x = 0$ and hence get ...
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1answer
64 views

Subspace in $I-T$ for bounded linear maps

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map.Show that the range of $I - T$ contains the subspace $$Y_T = \{x \in X: \limsup_{n\rightarrow \infty} n^2\|T^nx\| < ...
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2answers
563 views

The transpose in Banach spaces is bounded below

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map. Show that: If $T$ is surrjective then its transpose $T':X' \rightarrow X'$ is bounded below. My try: We know that ...
4
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339 views

Bounded linear maps in Banach spaces

Let $X$ be a Banach space and let $M: X \rightarrow X$ be a linear map. Prov that M is bounded iff there exists a set $S \subset X'$, dense in X', such that for each $\ell \in S$ the functional $m_l$ ...
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0answers
36 views

Extension of Linear Operatos [duplicate]

Possible Duplicate: Do continuous linear functions between Banach spaces extend? Is there an example of a pair of Banach spaces $X$ and $Y$, a subspace $E\subseteq X$ and a bounded linear ...
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2answers
809 views

Show that a finite-dimensional Banach space has a bijective compact operator

It is clear that if $ T: X \rightarrow X $ is a bijective compact operator, where $ X $ is a Banach space, then $ \dim(\text{Range}(T)) = \dim(X) $, which implies that $ \dim(X) $ must be $ < ...
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1answer
150 views

Weak compactness

Define a map $\varphi \colon [0,1]\to C[0,1]^*$ by $\varphi(x) = \delta_x$. Then $\varphi$ is a homeomorphism for the w*-topology. Let $K$ denote the image of $\varphi$. I have two questions: 1) Is ...
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Banach limit and its commutative counterpart, what do they tell us?

A Banach limit is a continuous linear functional $\Lambda$ on $\ell^{\infty}(\mathbb{N})$ satisfying: $\|\Lambda\|=\Lambda(1,1,1,\cdots)=1$; and ...
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Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
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1answer
263 views

Operators bounded below

Can one give me an easy example of an operator $T$ on a Banach space which is injective and has closed range and such that $\|T^2\|\neq \|T\|^2$?
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1answer
70 views

Prove $B_{m+n}=B_m * B_n$ if $B_k$ is the sigma-algebra of all Borel sets in $\mathbb R^k$

Let $B_k$ be the sigma-algebra of all Borel sets in $\mathbb R^k$. How can we prove that $B_{m+n}=B_m * B_n$? I am a beginner in analysis, hope to seek help here.
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3answers
351 views

Compact integral and multiplication operator in Banach spaces

Let $ A\colon C[0,1] \to C[0,1] $ $$ A(x)(t) = f(t)x(t) + \int_0^t x(s)ds,\quad f \in C[0,1]: f(1) \neq 0, \forall t \in [0,1] $$ Is $A$ a compact operator or not?
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166 views

Analogue of $S^{\bot \bot} = \overline{S}$ for Banach spaces which aren't Hilbert spaces

Let $X$ be a Banach space over the complex field. Let $X^*$ denote its topological dual. If $S$ is a subspace of $X$, write $$\mathrm{ann}_L(S)= \{ \varphi \in X^* : \varphi S = 0\}$$ and note the ...
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79 views

Spectral radius for linear compact maps

Prove or disprove the following assertions for a linear map $C$ from a Banach space $X$ into itself: a) If C is compact then its spectral radius equals the maximum of the absolute value of $C$ I'm ...
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2answers
2k views

Hilbert Space is reflexive

A normed space $X$ is reflexive iff $X^{**}=\{g_x:x\in X\}$ where $g_x$ is bounded linear functional on $X^*$ defined by $g_x(f)=f(x)$ for any $f\in X^*$. Let $X$ be a Hilbert space, would you help ...
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380 views

Continuity criterions for linear maps between Banach spaces?

I just did one exercise stating: Prove that the linear map $M: X \rightarrow C([0,1])$, is continuous iff for every $t\in[0,1]$, the rule $x\rightarrow (Mx)(t)$ defines a continuous linear functional ...
2
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1answer
456 views

The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
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1answer
132 views

Gateaux Differentiation in Infinite Dimensional Space

Let $X, Y$ be Banach spaces. A mapping $F: X\rightarrow Y$ is said to be Gateaux differentiable at $x_0\in X$ iff there exists a continuous linear mapping $A: X\rightarrow Y$ such that $$ \textbf{(*)} ...
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$(\lambda-a)^{-1}$ as limits of 'polynomials'

For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives \begin{equation} \sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a) \end{equation} for any unital $C^*$-subalgebra $\mathcal{B}$. ...
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2answers
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If $Lat(\mathcal{A})$ is trivial then $\mathcal{A}'$ consists of scalars.

This is related to Exercise 3 of Section 2.5 of Arveson's book on spectral theory. For those who are interested, we were asked to show the following $\mathcal{A}$ is a Banach *-algebra. ...
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Supremum of Banach Spaces

Let $X$ be a linear space with a family of complete norms $(\| \circ \|_I)_{I \in \mathcal{I}}$ on $X$, i.e. for every $I \in \mathcal{I}$ the tuple $(X,\|\circ\|_I)$ is a Banach space. Now define ...
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112 views

Show that the space $l^2= \{ a \in \mathbb{R}^\mathbb{N} | \sum |a_k|^{2} < \infty \}$ is a Banach space

If we have $l^2= \{ a \in \mathbb{R}^{\mathbb{N}} | \sum_{k=0}^{\infty} |a_k|^{2} < \infty \}$ and $||a||_2 = (\sum_{k=0}^{\infty} |a_k|^2 )^{1/2}$ 1 proposition: $l^2 $ is a vector ...
2
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1answer
168 views

Closable operator with finite range is continuous

I'm stuck on kind of a silly functional analysis problem. Suppose that $T:D(T) \subset X \to Y$ is a closable linear operator, where $X$ and $Y$ are Banach spaces and $R(T)$ is finite dimensional. I ...
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1answer
81 views

Let $X$ be a Banach space. Is $X$ a closed subspace of $X''$?

I'm trying to prove that $X$ is closed in $X''$, where $X$ is a Banach space. I know that $X$ is embeddable in $X''$. If the isomorphism was bijective, I could show that $X$ is closed since $X''$ ...
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A few questions about the Hilbert triple/Gelfand triple

I am attempting to fully understand Hilbert triples by reading Brezis' Function Analysis book. Consider $V \subset H \subset V^*$, where $V$ is Banach and $H$ is Hilbert. $V$ is dense in $H$. ...
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Space of bounded functions is reflexive if the domain is finite

Let $C_b(X)$ be a space of bounded continuous functions on a locally compact space $X$ equipped with the supremum norm. How to show that $C_b(X)$ is reflexive if and only if $X$ is finite?
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160 views

Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space

Let $X$ be a Banach space. Let $X^*$ denote the dual space . Would you help me, How to show that $(X^*)^{**}=(X^{**})^*$?
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Banach spaces $X$ for which the subspace of adjoint operators has finite codimension in $X^*$

For a Banach space $X$, let $A \subseteq \mathcal{B}(X^*)$ denote the closed subspace of those operators $T : X^* \to X^*$ such that $T = S^*$ for some $S \in \mathcal{B}(X)$, that is, $T$ is the ...
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Existence of a transpose of a linear operator

Let $X,Y$ be two linear spaces and let $b:X\times Y\to\Bbb R$ be a bilinear map. For any linear operator $A:X\to X$ we can define its $b$-transpose acting on $Y$ by the system of the following ...
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240 views

$\sigma(x)$ has no hole in the algebra of polynomials

Let $A$ be a unital banach algebra generated by two elements $1$ and $x$. Then it seems $\sigma(x)$ cannot have holes. At least this is true in the case for disk algebras. This amounts to prove that ...
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2answers
185 views

Bounded functionals on Banach spaces.

Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], f\rightarrow f(r)$ defines a bounded linear functional on $X$. Prove that there exists a ...
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1answer
629 views

Strictly convex Inequality in $l^p$

Let $1<p<\infty$. Let $x,y\in l^p$ such that $||x||_p=1$, $||y||_p=1$ and $x\neq y$. Would you help me to show that for any $0<t<1$, $||tx+(1-t)y||_p<1$. My answer : By using ...
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1answer
84 views

Does any closed subset in a Banach space have (at least one) point that has a minimum norm?

Does any closed subset in a Banach space have (at least one) point that has a minimum norm? I think this statement is obviously true, but how do I prove its correctness?
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1answer
850 views

Weakly Cauchy sequences need not be weakly convergent

A sequence $(x_n)$ in a Banach space $X$ is called weakly Cauchy if for every $\ell \in X'$ the sequence $(\ell(x_n))$ is Cauchy in the scalar field. I want to show that weakly Cauchy sequences ...
3
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0answers
191 views

Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...