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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Dual space of Bochner space

Let $B$ be a refexive Banach space. I want to show that $$(L^2(0,T;B))^* = L^2(0,T;B^*)$$ and that the dual pairing is $$\langle F,f \rangle_{L^2(0,T;B^*), L^2(0,T;B)} = \int_0^T \langle F(t), f(t) ...
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Turning Banach space into Banach algebra

Given a Banach space, how can we determine if we can turn it into a Banach algebra or not?
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How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
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274 views

A sequence converging weakly in $\ell^p$, for $p >1$ and failing to converge weakly for $p=1$

For $1 \le p < \infty$ and each index $n$, let $e_n \in \ell^p$ have $n$-th component 1 and all other componenets $0$. I want to show that $p>1 \Rightarrow \{e_n\} \to 0$ weakly in $\ell^p$ and ...
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A question about convergence in $L^p$. [duplicate]

Let $E$ be measurable and $1 \le p \le \infty$. Suppose $\{f_n\}_{n \in \mathbb{N}}$ all measurable and $\{f_n\}_{n \in \mathbb{N}} \to f$ pointwise a.e. $E$. For $p$ as above, I want to show that: ...
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1answer
102 views

Hypervolume of a $N$-dimensional ball in $p$-norm

Suppose I have a N-dimensional ball with radius R in p-norm: $$ \sum_{n=1}^N |x_n|^p = R^p $$ Is there a closed formula for its (hyper)volume? I can't find anything. If there isn't, can we at least ...
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1answer
118 views

Clarification of Reed and Simon proof of the open mapping theorem

I was reading Reed and Simon Methods of Mathematical Physics Volume 1 and have a question about a small their proof of the open mapping theorem for Banach spaces. Let $T:X\rightarrow Y$ be a bounded ...
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Is duality an exact functor on Banach spaces or Hilbert spaces?

Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence $0\to V'\to V\to V'' \to 0$, and ...
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On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
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471 views

The maximal ideal space of a Banach algebra

Let $G = \mathbb Z^n$, the fi nite cyclic group of order $n$, and let $A = l ^1(G)$, a Banach algebra over $\mathbb C$ when the product is convolution, de fined for $f, g \in A$ by $f * g(x) =\sum ...
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93 views

Extending definition of weighted $L^2$ norm

Is there a simple characterization of the domain of the semi-norm $\| \nabla (g \ast x) \|_{L^2 ( \mathbb{R}^3)}$, where $g$ is a gaussian convolution kernel? It is finite on $L^2$, but probably on a ...
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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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695 views

Isomorphism of Banach space

If $T:H\to B$ is isomorphism of Banach spaces and $H$ is Hilbert, must $B$ necessarily be Hilbert?
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71 views

How to show that these spaces are closed and complemented?

Suppose $X$ is a Banach space and $X_0$,$X_1$ are closed complemented subspaces of $X$. Let $P:X\rightarrow X_0$ and $I-P:X\rightarrow X_1$ ($I$ is the identity operator) be the projections ...
0
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1answer
46 views

A limit superior question in the context of the Neumann series

I'm trying to understand a step in the proof that the Neumann series converges: Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
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820 views

Rademacher function and weak convergence

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
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1answer
46 views

strange convergence in norm

I find it a bit strange that if $x_n \to x$ in Banach space $X$, then $|x_n|_X \to |x|_X$ by inverse triangle inequality... Surely that can't be right. Am I correct in this? Does this have a name? ...
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Parallelogram law valid in banach spaces?

It is known that the parallelogram law $\|x-y\|^2+\|x+y\|^2 = 2(\|x\|^2 + \|y\|^2)$ holds in any space with an inner product (the norm being induced by this inner product). Is this formula valid in ...
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1answer
994 views

If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive? To prove the contrapositive, it will suffice to assume that $X$ ...
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Literature suggestion for “strong convexity”

Does anyone know of a reference that discusses strong convexity and strong smoothness of proper convex functions over Banach spaces? All the references I find only deal with the finite dimensional ...
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1answer
802 views

There are compact operators that are not norm-limits of finite-rank operators

Given an example of a Banach space for which There are compact operators that are not norm-limits of finite-rank operators. Tanks for answer
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1answer
56 views

there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
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On every infinite-dimensional Banach space there exists a discontinuous linear functional.

On every infinite-dimensional Banach space there exists a discontinuous linear functional. Assuming the axiom of choice, every vector space has a basis. With an infinite basis, I can define on a ...
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1answer
362 views

Continuous Coercive Functional not Bounded Below

Let $X$ be a infinite dimensional Banach space. How to construct a example of a continuous function $f:X\rightarrow\mathbb{R}$ such that $f$ is coercive, but is not bounded below. $f$ is coercive if ...
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Mid-Point Convexity Implies Convexity in Banach Spaces? [duplicate]

Possible Duplicate: Showing that $f$ is convex Let $X$ be a real Banach space and $f:X\rightarrow \mathbb{R}$ a continuous function. We say that $f$ is Mid-Point convex if for all $x,y\in ...
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How to construct an “explicit” element of $(\ell^\infty(\mathbb N))^* \setminus \ell^1(\mathbb N)$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? Everything is in the title: How to ...
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Is it a Banach space? If so what is its dual?

Let $(E_n)$ be a sequence of Banach spaces and $(w_n)$ be a sequence of positive real numbers. For $1\leq p <\infty$ define $\bigoplus\limits_P E_n:=\{(x_n):x_n\in E_n,\sum\limits_n\lVert ...
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Operators bounded below on a linearly dense subset

A bounded operator $T\colon X\to Y$ is bounded below if there is $\lambda>0$ such that $\|Tx\|\geqslant \lambda \|x\|$ for all $x\in X$. Suppose $D$ is a linearly dense subset of $X$ such that ...
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Banach dual space integral

Let $X$ be a Banach space and $f_t \in X^*$ for each $t \in [0,t_0]$. Suppose that $$\int_0^{t_0} f_t(x) = 0$$ for all $x \in X$. 1) Does it make sense to write $\int_0^{t_0}f_t = 0$? 2) If so, does ...
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1answer
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Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?

Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
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Homeomorphisms on X and automorphisms on C(X)

Let $ X $ be a compact Hausdorff space. Let $ \psi $ be a homeomorphism on $ X $. Let $ \text{Aut}(C(X)) $ be the group of automorphisms of $ C(X) $, and $ \text{Homeo}(X) $ be the group of ...
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Question on continuity of functions from $X\times X\rightarrow Y$

I am stuck on the following problem, which I do not believe to be so difficult. Let $X$ and $Y$ be Banach spaces. Let $f:X\times X\rightarrow Y$ be a function such that for any fixed $x_0$, ...
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How to define Continuous $f: B\rightarrow B$ without fixed points?

Let $X$ be a infinite dimensional Banach space and $B=\{x\in X:\ \|x\|\leq 1\}$. How to define a continuous $f:B\rightarrow B$ without fixed points? Edit 1: I have changed the question a litlle, ...
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A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other

The functionals $$ \phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t $$ define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$. a) Show that $(\phi_n)$ converges ...
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Example of atlas for sequence space

Is it possible to construct and atlas for space $\ell_1$? I.e. give an example of collection $(U_\alpha,\phi_\alpha)$, such that $\cup U_\alpha=\ell_1$ and $\phi_\alpha:U_\alpha\to R^{n}$ is a ...
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1answer
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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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1answer
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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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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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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 ...
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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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1answer
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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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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 ...
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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 ...
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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
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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 ...
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1answer
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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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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
111 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: ...