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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Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal ...
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Prove or disprove that T:[0,2π] -> [0,2π] given by Tx = sin(2014x) is a contraction

i know that if we assume $T:[a,b] \to [a,b] $ and if $|T'(x)| ≤ α \space \forall \space a≤x≤b$ then T is a contraction . but unsure of how to apply that to this question
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100 views

When is the dual ball of $L_1(\mu)$ weak*-sequentially compact?

Where could I find a direct proof showing that the dual ball of $L_1(\mu)$ is weak*-sequentially compact? Since $(L_1(\mu))^*=L_\infty(\mu)$, I mean the unit ball $B_{L_\infty(\mu)}$ of ...
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88 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
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28 views

Coanalytic families of Banach spaces

Is it true that if $G$ is a coanalytic family of separable Banach spaces, which is not Borel, and $H\subset G$ is not Borel, then H is coanalytic? This is something I have come across. I am reading a ...
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34 views

if $f$ is in Banach space, then $\nabla f $ is in the dual space?

I am not very deep in advanced real analysis. Could you help me decipher the following two phrases hold? 1) if $f$ is in Banach space $\mathcal{B}$, then $\nabla f $ is in the dual space ...
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32 views

Krein Milman Property

If a closed bounded (not compact) set $X$ in a Banach space $B$ (like $L^1$) has extreme point(s), must the max of a linear functional defined on $X$ occur at one of them? I suppose it depends on $B$. ...
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29 views

Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
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29 views

the second derivative for a composite function

Let $f: U\rightarrow \mathbb{R}$ where $U\subseteq X$ a normed vector space and suppose $f(\mathbf{u})\in C^{2}$.Let $\mathbf{x}\in U$ and let $r>0$ be such that $B(\mathbf{x},r)\subseteq U. $ ...
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47 views

How to prove a space is a dual space?

How does one go about proving that a space is a dual space? The only thing I can think of is to prove that the space is isomorphic to a dual space. Is there a better way to do this? Thank you
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44 views

A paradox derived from the open mapping theorem

The problem comes from Erwin Kreyszig's Introductory Functional Analysis with Applications, section 7.4, problem 4: Let $T:l^2\mapsto l^2$ be defined by $y=Tx, x=(\xi_j), y=(\eta_j), ...
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61 views

Proving completeness of $L^p$

I want to make sure my understanding of the proof is correct. For a Cauchy sequence $\{f_n\}$ in $L^p$, we want to find a $f\in L^p$ such that $f_n\stackrel{L^p}\to f$ Now, skipping the ...
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1answer
27 views

A question about regulated functions

Let $X$ be a Banach space and consider the following definition. Definition. $f:[a,b]\to X$ is regulated if it has one-sided limits at every point of $[a,b]$, i.e. for every $c\in [a,b)$ there is a ...
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Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”? [closed]

This is true for finite-dimensional spaces, of course. To be precise, let $T$ be an operator on a complex Banach space $X$ which is not finite-dimensional. For each $\lambda \in \mathbb{C}$, let ...
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Isomorphism on dense subset

I am wondering if the following could be done. I want to show two Banach spaces $X$ and $Y$ are isomorphic. If $A$ is dense in $X$, and $B$ is dense in $Y$, is it sufficient to show there is an ...
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73 views

Separable spaces isomorphic

I am reading a proof and it states the following without proof: Two separable Banach spaces $X$ and $Y$ are isomorphic iff there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ such that ...
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42 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwarz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
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54 views

Examples of Sobolev Spaces

I would like to apologize in advance for this trivial question! Does constant functions $u \equiv C$ and, in partucular, $u \equiv 0$ belong to $W_{0}^{1,2}(\Omega)$? Update 2: $\Omega \subset ...
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1answer
69 views

closed, convex, absorbing subset of a banach space

There is a nice theorem that every closed, convex, absorbing subset of a banach space includes an open ball arround $0$. Can you give an example where the theorem fails if we do not assume the subset ...
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1answer
37 views

spectral theory expandable to arbitrary polynomials?

Given a Banach space $X$ and closed operators $A_i$ ($i \in \left\{0,...,n\right\}$) which have a common domain $D$ that is dense in $X$. An obvious candidate for the title of "generalised resolvent ...
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241 views

Does Continuity in Weak Operator Topology imply Continuity in Strong Operator Topology?

This homework problem has puzzled me for almost a year. As nobody in the class has figured it out, I would like to seek a proof or a disproof here. Problem (Prove or Disprove) Let $X$ be a Banach ...
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34 views

Are continuous bounded functions a subspace of $L^2$?

I have a problem where I need to work with functions that are square-integrable, bounded and continuous, i.e. the space $ L^2 \supset X = \left\{ f \in L^2 \mid f \text{ bounded, continuous}\right\} ...
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38 views

(Riesz's lemma) A closed subspace of a Banach space

Let V be a Banach space over R Let W be a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ And my proof ...
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32 views

Banach space and its closed subspace. a vectors satisfying inequality.

V=a Banach space over R W=a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ I have shown that there exists ...
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23 views

Existence of a vector whose norm is 1 in a Banach space

Given a finite dimensional Banach space V over reals, I have to show that there exists $v \in V$ such that $\|v\|=1$ At first, I thought that there's an identity element I in V. And $\|I\|=1$. But ...
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24 views

Projection on closed subspace of $L^1$, $L^{\infty}$

For $p=1,\infty$ let $K$ be a closed subspace of $L^p(\mathbb{R},m)$. According to this question, it should be easy to find examples of $K$ and $f\in L^p(\mathbb{R},m)$ such that there exists a ...
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58 views

Projection on closed subspace of $L^p$, $1<p<\infty$

Let $1<p<\infty$ and $K$ be a closed subspace of $L^p(X, \mathcal{M}, \mu)$. If $f\in L^p$ then there exists a unique $h\in K$ such that $||f-h||_p$ equals $$ \text{dist}(f,K)=\inf_{g\in ...
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37 views

Is this a Hilbert space?

For $n\geq 2$, we let $\mathcal{H}$ be the complex vector space of all complex-valued functions on $[0,1]$ such that (a) $f(0)=0$, (b) for $1\leq k\leq n-1$, $f^{(k)}$ exists everywhere and is ...
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55 views

Can the ball $B(0,r_0)$ be covered with a finite number of balls of radius $<r_0$

Consider an infinite dimensional Banach space $X$. Let $B(0,r_0)$ be the ball with radius $r_0$. We know that the ball $B(0,r_0)$ is not relatively compact, so it is not totally bounded. This implies ...
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Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
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Banach space of p-Lipschitz functions

Given $p\in\mathbb{R}$, consider the space: $$ Lip(p) = \left\{f:[0,1] \longrightarrow \mathbb{R} : \mbox{ $f$ is $p$-Lipschitz} \right\}$$ i.e.: there is $M>0$ such that ...
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Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
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69 views

What is abstraction of direction in considering vectors such as used in Engineering & Physics?

In the use of vectors of engineering and physics, we encounter objects that obey the axioms of a vector space but also have two new attributes of length (or, magnitude) and direction (e.g. direction ...
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39 views

is the complexification of a finitely strictly singular operator itself FSS?

Let $X$ and $Y$ be real Banach spaces, and let $X_\mathbb{C}$ and $Y_\mathbb{C}$ denote their respective complexifications. Suppose $T:X\to Y$ is a bounded linear operator which is finitely strictly ...
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44 views

Gateaux and Frechet derivatives on $\mathbb{R}^2$.

I have the following problem: Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by: \begin{equation} f(x,y)= \frac{x^3y}{x^4+y^2}, \quad x \neq 0, y\neq 0 \end{equation} and: \begin{equation} ...
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62 views

Hamel basis and Banach spaces

Suppose $X$ is a linear space and $X$ has a Hamel basis with uncountable number of elements. Does there exist a norm on $X$ such that $X$ is a Banach space with respect to this norm?
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31 views

Bijective bounded linear operator is invertible

The following is an exercise from Halmos book "A Hilbert space problem book" : Exercise: If $H$ and $K$ are Hilbert spaces, and if $A$ is a bounded linear transformation that maps $H$ one to one and ...
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Definition of Szlenk index, $w*$ closed set.

I am reading a paper and it has the following: Let $X$ be a separable Banach space. Given $\epsilon>0$, and a $w^*$- closed subset $P$ of $B_{X^*}$, we let $P_\epsilon'=\{x^*\in P \mid $ for all ...
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32 views

Do any $\ell^{p}(\omega)$ have the extension property?

Definition 1. A metric space (normed space) $X$ has the extension property exactly in case, for all finite $A\subseteq X$ and isometric (linear isometric) $f:A\rightarrow X$, there exists an extension ...
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18 views

Verification of a fact, separable Banach spaces, closed subset.

I am reading a proof and it says to verify the following: Suppose $Z$ is a separable Banach space and $F$ is a closed subset of $Z$. Let $\mathcal{O}$ be a countable basis of open subsets of $Z$. We ...
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strong epis in the category of banach spaces with linear contractions

In Borceux's Handbook volume 1, page 145, the strong epis in the category of Banach spaces with bounded linear maps of norm <= 1 is characterized as the maps whose restriction on the unit balls is ...
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50 views

Lp spaces - example of function

Can you give me an example of function which: $$f \in L^{p}[a,b]$$ but $$f \not\in L^{\infty}[a,b]$$ $L^{\infty}[a,b]$ is space of essentially bounded function at interval $[a,b]$ $1 \le p < ...
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Prove that $S:F \to \mathcal{L}(E,F)$ is a topological isomorphism

Let $E$ and $F$ be normed spaces, $E \neq \{ 0 \}$ and $x_0 \in E \backslash \{ 0 \}$, $x_0 \in E'$ such that $x_0'(x_0)=1$. Prove that the function $S: F \to \mathcal{L}(E,F)$, $S(y)=T_y$ defined ...
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40 views

Compact metric implies uncountable w*-dense set

I am reading a proof of the following: Let $X$ be a separable Banach space. The Szlenk index is countable iff $X*$ is separable. In the proof of => it uses the following: If $X^*$ is not separable, ...
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46 views

question about the proof that the set C[a,b] with uniform norm is complete

I am trying to understand the proof that the set of continuous function is complete under uniform/supremum norm. First, suppose we have a Cauchy sequence of continuous functions ${f_n(t)}$ with ...
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60 views

A question in Banach space

Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively. ...
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167 views

Converse of uniform boundedness principle

The uniform boundedness principle says if we have a collection of bounded linear operators $\Gamma$ from a banach space $X$ into a normed vector space $Y$, which is pointwise bounded on $X$, i.e. ...
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72 views

Weak convergence + compactness = strong convergence? [duplicate]

Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for ...
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1answer
32 views

Bijection between a Hilbert and a Banach space

I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to ...
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1answer
79 views

The Banach–Mazur distance for finite-dimensional $\ell_p$

Let $\ell_p$ denote the usual infinite-dimensional sequence space, and if $n\in\mathbb{Z}^+$ then we let $\ell_p^n$ denote its $n$-dimensional counterpart. Conjecture. Let $1\leq p<\infty$. ...