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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Proof completeness of $L^p$

I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$). I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then ...
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27 views

Why $L^1$ is not reflexive [duplicate]

We already known that $$ (L^p(\Omega))^* = L^q(\Omega), $$ for all $1\le p < \infty $ and $q$ is the exponent conjugate to $p$. So that, $L^p(\Omega)$ is reflexive with $1<p<\infty$. However, ...
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22 views

$Y\subset X^\ast$ separable implies $Y\hookrightarrow Z^\ast$ for some separable subspace $Z\subset X$

Let $X$ be a normed space and $Y$ be a separable subspace of $X^\ast$. I'd like to show that there is a separable subspace $Z\subset X$ s.t. Y isometrically embeds in $Z^\ast$. Has anyone a hint what ...
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23 views

Given $X,Y$ are Banach spaces with norms $\|x\|_X,\|y\|_Y$, prove $\|(x,y)\|=\max(\|x\|_X,\|y\|_Y)$ is a norm and defines a Banach space

Here is the question: Let $X$ and $Y$ be Banach spaces with norms $\|x\|_X$ and $\|y\|_Y$ respectively. Prove that $$\|(x,y)\|=\max\{\|x\|_X,\|y\|_Y\}$$ defines a norm on $X\times Y$, and that ...
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22 views

Find $f$ such that the contraction $\phi$ has a fixed-point $\rho= \sqrt{2}$

I use the Newton method and the Banach fixed-point theorem and have: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous and $f: I \rightarrow \mathbb{R}$ a ...
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17 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
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29 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
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47 views

$p$-summable series in a Banach space

Let $E$ be a Banach space and denote its dual space by $E^*$. Let $p \in [1, \infty)$ and $x : \mathbb{N}\rightarrow E$ be such that for every $\phi \in E^*$, $$\left( \sum_{n=1}^{\infty} \lvert ...
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10 views

Convergence with respect to the graph norm

Let $T:A\subset E\to F$ be a linear closed operator with dense domain. (E,F are Banach spaces) and $x_n$ a sequence in A such that $||x_n-x||_E\to 0$. Let $||x||_1:=||x||_E+||T(x)||_F$ be the graph ...
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$L^{p}$ spaces and their properties

I have aquestion :Idont know how to show that if $1<p<q<\infty$ , then $L^{q}$(0,1)$\subset$$L^{p}$(0,1) and $\mid\mid f\mid\mid$$_p$ < $\mid\mid f\mid\mid$$_q$ ,f $\in$$L^{q}$(0,1)? ...
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Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
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20 views

Newton method and the Banach fixed-point theorem

I try to combine the Newton method and the Banach fixed-point theorem but I have still some questions: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. ...
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1answer
18 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core ...
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20 views

Natural *-isomorphism

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$. My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that ...
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16 views

Gelfand triple: what happens if we don't identify the pivot Hilbert space with its dual?

People usually say $V \subset H = H^* \subset V^*$ is a Gelfand triple if $V$ is continuously and densely embedded in $H$ and $H$ is identified with its dual. Sometimes they do not mentioned that ...
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67 views

Banach fixed point theorem (application)

Let $X$ and $Y$ be Banach Spaces, and $T: X \to Y$, where T is continuous, linear and bijective, and let $S: X \to Y$ (where $S$ is continuous and linear) with $|S|\cdot|T^{-1}|< 1$. Show that ...
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1answer
93 views

What's Helly's theorem in the proof of the Goldstine–Weston density theorem

I have a problem in understanding a proof of Goldstine–Weston density theorem. The only thing I don't know in the proof is the part of Helly's theorem to be related. The Goldstine–Weston density ...
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1answer
27 views

Positive linear functional on an involutive Banach algebra

Why is every positive linear functional on an involutive Banach algebra with a bounded approximate continuous?
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27 views

Convergence in $L^1 \cap L^2$

I am very confused about the following: Assume we have a sequence of functions $f_n \in$$L^1 \cap L^2 (\mathbb{R}^n)$. Then is it true that if this sequence is Cauchy both in $L^1$ and $L^2$, two ...
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20 views

Nonequivalent norms on infinite-dimensional linear space

I've just proven that for every infinite-dimensional space with a norm $(V, ||~||)$ we can find a discontinuous linear functional $ \phi $. But next I'm trying to prove the following: The norm $ ...
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28 views

Banach space, dual of its quotient, isometric isomorphism

I'm trying to prove the following: If $X$ is a Banach space, $Y$ its closed linear subspace, $\pi : X \rightarrow \ ^X/_Y$ is a canonical projection, then $ (^X/_Y)^* \ni u \rightarrow u \circ \pi ...
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35 views

Space of real polynomials of one variable isn't complete

Consider $E$ - the vector space of all real polynomials of one variable. I need to prove that it is not complete under any norm. I was thinking I could use the fact that certain functions, for ...
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118 views

Why isn't $C[0,1]$ a Banach space in this unusual norm?

I need to answer the following question: Let $X$ be the normed space $X=C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't this a Banach space?
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Separable dual space implies existence of complete series

Let $\{x_n\}$ be a basis for a Banach space $X$ and let $\{f_n\}$ be the associated sequence of coefficient functionals. Prove or disprove: if $X^\ast$ is separable, then ${f_n}$ is complete ...
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104 views

Is $\mathcal{B}(H)$ complemented in $\ell_\infty(I, H)$

Let $H$ be an infinite diensional Hilbert space. Consider unit ball of $H$ as index set, denote it by $I$, then we have an isometric embedding $$ ...
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27 views

Neighborhood base of weak topology

If $X$ is a Banach space, the weak topology on $X$ is the weakest topology in which each functional $f$ in $X^\ast$ is continuous. I have some difficulties in understanding its neighborhood basis in ...
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49 views

What is the image of operator exponential?

Given a Banach space $V$ and a bounded linear operator $A:V\to V$, the operator $e^A$ is bounded and invertible. When $V$ is finite dimensional, every invertible operator is of the form $e^B$ (one can ...
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38 views

Show $\ell_\infty (M)$ is a Banach Space

I'm working on problems from Carothers' Real Analysis. The following problem is in the section on completions. Given any metric space $(M,d)$, check that $\ell_\infty(M)$ is a Banach space. ...
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Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?

It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
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47 views

Prove $\dim X = \mathfrak{c}$ for every infinite dimentional Banach space

Let $X$ be an infinite-dimensional Separable Banach Space. Prove that $\dim X=\mathfrak{c}$. On the direction of $\dim X\ge \mathfrak{c}$, I thought taking the subset of all elements $x\in X$ ...
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26 views

There is no continuous function h on unit circle such that u=exp ih when spectrum u is entire unit circle

Let $\Gamma$ be the unit circle. Let u be the unitary element in $C(\Gamma)$ defined by $u(\lambda)=\lambda$. Show that there is no continuous function h on unite circle such that u=exp ih. ...
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functional calculus on a set of normal elements is continuous

Let $K$ be a compact subset of $\Bbb C$. Let $A_K$ denote the set of all normal elements $x$ with $\sigma_A(x)\subset K$. If $f$ is a continuous function on $K$, then the functional calculus :$x\in ...
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Can a linear functional be infinite at a point?

On a Banach (or Hilbert) space $X$, when we define a linear functional (not necessarily bounded), we define it to be a linear function from the elements of $X$ to the field $\Bbb F$. (Say, $\Bbb R$). ...
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Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

the following is a proposition: If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism. To prove it, the author ...
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Spectral radius of a normal element in a Banach algebra

I know that if $A$ is a C*-algebra, then $||x||=||x||_{sp}$ for every normal element $x\in A$. I like to find an example of a normal element $x$ in a involutive Banach algebra with $||x||\neq ...
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$(C_{[0;1]}, \Vert . \Vert_1)$ is not a Banach space

I'm going to prove that $C_{[0;1]}$ is not a Banach space w.r.s to the norm $\Vert x \Vert_1 = \int_{0}^{1} |x(t)| dt$ by consider the series $\sum_{n=1}^{\infty}x_n$ where $x_{n}(t)= t^{n} \cdot ...
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44 views

Spectrum of an element of a non unital C*-algebra

I know that spectrum of an element $x$ of a unital C*-algebra is nonempty. I like to find an example of a non unital C*-algebra that has an element with empty spectrum, if it exists. Motivation I ...
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Spectral radius of an element in a C*-algebra

The following is an proposition of Takesaki's Operator Theory: For any element $x$ of a Banach algebra ${\cal A}$, we have $$||x||_{sp}=\lim_{n\to \infty}||x^n||^{\frac{1}{n}}$$ Proof: My ...
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21 views

Book/Papers for properties of convex/ uniformly convex Banach Spaces

I am looking for reference books and research articles which cover analysis of uniformly convex and strictly convex Banach spaces.
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46 views

Uniqueness of projection in a Banach space

Let $X$ be a Banach space, $M$ be a subspace of $X$ and $x \in X$ be any vector in $X$. Consider $\displaystyle \hat{x}_M=\arg \inf_{m\in M}\|x - m\|$. Under what conditions for $l_p$ norms $p = ...
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Minimum distance from a subspace in a Banach space

$X,Y,Z$ are disjoint sets of vectors in a Banach space M. $S_X,S_Y,S_Z$ denotes the subspace formed by the vectors in $X, Y$ and $Z$ respectively. $S_{YZ}$ and $S_{XZ}$ denote the subspaces formed by ...
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54 views

Two questions on Banach function spaces

I have recently started studying Banach function spaces over $\sigma$-finite measure spaces. By a Banach function space I mean: Let $\left(R, \mu \right)$ be a $\sigma$-finite measure space and let ...
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35 views

Proving uniformly convergence on a Banach Space

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$ and $$ {\cal L}_0^2(\mathbb R)=\left\{f:\mathbb R\mapsto\mathbb R\ |\ ...
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Does Hilbert space intuition carry over to Banach space?

Given a Banach space $X$ and its dual space $X^*$. Suppose $a$ and $b$ are two unit norm vectors in $X$. $a^*$ and $b^*$ be unit norm elements in dual space $X^*$ such that $a$ and $a^*$ as well as ...
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75 views

Properties of a $B^\ast$-algebra

Defining a complex Banach algebra as a $B^\ast$-algebra when it is equipped with an application $\ast:B\to B$ such that for any $x,y\in B$ $$(x+y)^\ast=x^\ast+y^\ast,\quad(xy)^\ast=x^\ast ...
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Is it true that $\dim(X) \leq \dim(X^{\ast})$ for every infinite dimentional banach space $X$?

So given an arbitrary infinite dimensional Banach space $X$ can we deduce that it's dimension $\dim(X)$ (the cardinality of one of it's Hamel bases) is less or equal of the dimension $\dim(X^{\ast})$ ...
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25 views

Convergence of sequence of functions on Banach space

Let $\{f_{\alpha_n}\}\subset{\cal L}_2^0(\mathbb R)$ be a sequence function converging to $g$ where ${\cal L}_2^0(\mathbb R)$ is a Banach space defined by $$ {\cal L}_2^0(\mathbb ...
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1answer
23 views

Norm of a character in a non-unital Banach algebra

Let $\cal A$ be an abelian non-unital Banach algebra and $h:{\cal A}\to {\Bbb C}$ be a homomorphism. If ${\cal A}$ has an approximate identity $\{e_i\}$ such that $||e_i||\leq 1$ for all i, then ...
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48 views

Complex Measures: Integration

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! Integration w.r.t. complex measure usually is defined via the Radon-Nikodym ...
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51 views

Spectral Measures: Integration

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...