Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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1
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2answers
111 views

A problem on the bounds of Lp-norms

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ...
0
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1answer
38 views

Is $D$ well-defined?

In my text there's a problem which reads as: Consider $C[0, 1]$ with the norm $\|.\|_\infty$. Let $Y$ be the vector subspace of all differentiable functions on $[0, 1].$ Consider the linear map ...
3
votes
1answer
60 views

Let $T:X\to Y$ be continuous at $0.$ Then $\exists~k>0$ such that $\|Tx\|<k\|x\|.$

Let $T:X\to Y,~(X,Y$ being Normed Linear Spaces$)$, be a linear transformation continuous at $0.$ Then $\exists~k>0$ such that $\|Tx\|<k\|x\|.$ My attempt: $T$ is continuous at $0\implies$ for ...
1
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0answers
133 views

Proof that normed space is Banach space

I have to prove that $(l^{\infty},\|\cdot\|_{\infty})$ is Banach space and I have some difficulties. This is what I've done. $l^\infty=\{x=\langle x_k\rangle, k\in N|\exists M>0 \ such\ ...
2
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1answer
54 views

$(P[0,1],\|\|_{\infty})$ be the norm linear space

Let $(P[0,1],\|\|_{\infty})$ be the norm linear space and $T$ be the differentiation operator on it. Then $1.$ $T$ is onto right? but NOT injective as $\ker T=\{\text{ all constants }\}$ $2.P[0,1]$ ...
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0answers
44 views

$E_1+E_2$ is open if both open?

if $X$ be a norm linear space and $E_1,E_2\subseteq X$ then $E_1+E_2=\{x+y:x\in E_1,y\in E_2\}$ is open if both open? is open if one is open and another is closed? closed if both are closed? I just ...
10
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1answer
215 views

Proof that multiplying by the scalar 1 does not change the vector in a normed vector space.

I'm beginning a self-study of functional analysis, and I seem to have come to a halt trying to solve the first problem in the first problem set, and was wondering if someone could give me a pointer. ...
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2answers
44 views

how to show $\|T\|\le 1$

Given that $M$ is a closed linear subspace of $N$ and if $T$ is a natural mapping of $N\to N/M:x\to x+M$, I have shown that $T$ is continuous , but I am not able to show $\|T\|\le 1$ Thank you for ...
2
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1answer
75 views

Example of infinite dimensional B* space where weak convergence does imply strong convergence

So I know that weak convergence does imply strong convergence if the dimension of the space is finite, and that in general it does not in infinite dimension. But I was wondering if there were any ...
1
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1answer
134 views

A distance-minimizing continuous projection onto a finite-dimensional subspace?

Let $E$ be a Banach space, which need not be a Hilbert space, and let $F$ be a finite-dimensional subspace of $E$. Suppose that for all $x \in E$, there exists a $y \in F$ realizing the minimal ...
3
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2answers
22 views

Density and the size of coefficients

Let $E$ be a Banach space, and $F$ a dense subspace spanned by a countable base $y_i$ of unit norm. Let $x \in E$ and $x_n = \sum_{i_n=1}^{N_n} a_{i_n} y_{i_n}$ be a sequence of elements of $E$ ...
3
votes
1answer
82 views

Do I have a Banach space given the following norm?

This is very very similar to my other question asked three months ago. That time there was no Banach space because an integral in the norm definition allowed a counter-example. Once again I have a ...
0
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1answer
47 views

$\mathbb C$ over the field $\mathbb R$ or $\mathbb C$ over the field $\mathbb C$

When we talk about the topology of the complex plane what type of $\mathbb C$ as a normed linear space we get concerned about viz. $\mathbb C$ over the field $\mathbb R$ or $\mathbb C$ over the ...
2
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2answers
86 views

Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
1
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1answer
142 views

How to show Zygmund space is Hölder space?

The motivation of this question is to show that Zygmund space is Hölder space, in certain cases. For simplicity, take $s\in (0,1)$, I want to show $$\|f\| = \|f\|_\infty + \sup_{x,y\in ...
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1answer
111 views

Showing a norm preserving isomorphism of vector spaces

Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
1
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2answers
59 views

how to show that $A_kB_k\to AB?$

Let in the space $M(n,\mathbb R)=$ set of all $n\times n$ real matrices endowned with $\| \cdot \|_2,~A_k\to A,~B_k\to B.$ Then how to show that $A_kB_k\to AB?$
2
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0answers
67 views

Distance from a point to a plane in normed spaces

How can we calculate the distance from a point to a plane in normed spaces ? where we are not inner product, for example: Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm ...
2
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1answer
279 views

How to proof homeomorphism between open ball and normic space

How can I prove that an open ball $B$ in a normed vector space $X$ is homeomorphic to $X$?
5
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1answer
544 views

Prove the boundedness of a bilinear continuous mapping.

Let $X,Y,Z$ are Banach spaces and $$B:X\times Y\to Z$$ is bilinear and continuous. Prove that there exists $M<\infty$ such that $$\lVert B(x,y)\rVert \leq M\lVert x\rVert\lVert y\rVert.$$ Is ...
1
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0answers
61 views

Definition of a projection on a normed space? Banach space?

Given a vector space $V$, a projection $V\to V$ is an idempotent linear map. For a normed space do we require anything else of the definition like continuity? Is the image required to be closed in ...
1
vote
1answer
37 views

Distance of a function from a subspace

Let $f \in L^2([-a,a])$. Trying to find $\mathrm{dist}(f,S)$ in $L^2([-a,a])$ (where S is the subspace of real polynomials of max degree $2$, like $a+bx+cx^2$) and knowing that $\langle f,a\rangle=0$ ...
0
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1answer
100 views

Determinant of Schur Complement

If I have an $n \times n$ real-valued non-symmetric matrix $\mathbf{M}$, which has determinant $|\mathbf{M}| > 0$, what can I say about the determinant of the matrix $\mathbf{Q}^T \mathbf{M}^{-1} ...
1
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2answers
67 views

Is't a correct observation that No norm on $B[0,1]$ can be found to make $C[0,1]$ open in it?

There's a problem in my text which reads as: Show that $C[0,1]$ is not an open subset of $(B[0,1],\|.\|_\infty).$ I've already shown in a previous example that for any open subspace $Y$ of a ...
3
votes
1answer
765 views

Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...
2
votes
1answer
104 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
2
votes
1answer
60 views

If $X$ is a normed space and $Y \subset X$, show $\max\limits_{\substack{f \in X^*,\\ \|f\|\leq 1,\,f|_{Y}=0\;}} |f(x)|=\inf\limits_{y \in Y}|x-y|$

Let $Y \subset X$ a subspace of normed space $X$. Show that $$\displaystyle \max_{f \in X^*, \ ||f||\leq 1, \ f|_{Y}=0} |f(x)|=\inf_{y \in Y}|x-y|.$$
4
votes
1answer
39 views

Let $(X, \|.\|)$ be an NLS, $x\in X$ and $0 < r<s.$ Show that $B(x, r)\subsetneq B (x, s).$

In my text I've found the problem: Let $(X, \|.\|)$ be an normed linear space, $x\in X$ and $0 < r<s.$ Show that $B(x, r)\subsetneq B (x, s).$ I can see if $\exists~y\in X-\{0\}$ then ...
3
votes
1answer
54 views

Comparing norms on $\mathbb{R}^n$

We know that $\mathbb{R}^n $ is normed linear space with respect to the norms defined as follows $\Vert x\Vert_{1} = \sum_{i =1}^n |x_i|$ $\Vert x\Vert_{2} = (\sum_{i =1}^n |x_i|^2)^{1/2 }$ $\Vert ...
0
votes
1answer
93 views

Questions regarding Holder's and Minkowski's inequality

I've some questions regarding Holder's and Minkowski's inequality as given in my text: Does the author consider the case $q=\infty$ in the equality case of lemma 1.1.36? Shouldn't the author ...
1
vote
1answer
214 views

Question over function twice differentiable if $D^2 f$ is constant

Let $E$ and $F$ be normed spaces. What can you say of a function $f:A\subseteq E\to F$ with $A$ open in $E$ twice differentiable, if $D^2 f$ is constant? This is a very open question that do not ...
0
votes
1answer
75 views

Bounded Derivate of a differentiable and Lipschitz function

Let $E, F$ normed spaces and $f:A\subseteq E\to F$ with $A$ open set, suppose that $f$ is differentiable at $a\in A$ and that $f$ is locally Lipschitz of constant $k>0$ in $a$. Show that ...
8
votes
3answers
208 views

Does $\|\cdot\|_2:C_\mathbb R([0,1],\mathbb C)\to\mathbb R:f\mapsto\sqrt{\int_0^1|f(t)|^2dt}$ come from any inner product?

I'm trying to show $\|\cdot\|_2$ is a norm on the $\mathbb C$-vector space $C([0,1],\mathbb C)$ where $$\|\cdot\|_2:C([0,1],\mathbb C)\to\mathbb R:f\mapsto\sqrt{\int_0^1|f(t)|^2dt}$$ I've stuck in ...
1
vote
1answer
126 views

Exercise of differentiable functions in $\mathcal{C}[0,1]$

Consider $E=\mathcal{C}[0,1]$ with norm $\|\cdot\|_\infty$. For which $x$ is differentiable the following functions: a) $f:E\rightarrow E$ defined by $f(x)(t)=|x(t)|^{2/3}$ b) $f:E\rightarrow ...
1
vote
1answer
44 views

Special operator on a normed space

Let $E$ be a normed space and $T \in L(E)$ with $\|Tx\|\lt\|x\|$ for all $x\ne0$ and $\|T\|=1$. I want to prove the following: $A=\{x\in E: \|Tx\|\ge1\}$ is closed. There is no $x\in A$ with ...
1
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1answer
112 views

Differentiability of the supremum norm in $\ell^{\infty}$

Let $\ell^{\infty}=\{x\in \mathbb{R}^{\mathbb{N}}: x\,\, \text{is bounded}\}$ and $E=\{x\in \ell^{\infty}:x_n\rightarrow 0\}$ with the norm $||\cdot||_{\infty}$ and let $f(x)=||x||_{\infty}$. How to ...
0
votes
1answer
76 views

Differentiable function exercise in $B(\mathbf{0},r)$

Let $E, F$ normed spaces and Suppose that $g:E\rightarrow F$ is differentiable in every point of $B(\mathbf{0},r)$, that $g(\mathbf0)=\mathbf0$, and that $\|Dg(x)\|\leq\lambda$ for all $x\in ...
1
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0answers
33 views

$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^\infty\lambda_i\cdot a_i:a_i\in A, \lambda_i\ge0,\sum_{i=1}^\infty\lambda_i=1\right\}$ is superconvex

Let $X$ be a Banach space and $A\subset X$ a subset bounded. Denote by $\operatorname{sconv}(A)$ the superconvex hull of $A$: $$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^{\infty}\lambda_i\cdot ...
1
vote
2answers
145 views

Do equivalent norms preserve dual spaces?

Suppose that $X^*$ is the dual space of a normed space $X$. If we renorm the space $X^*$ with a new norm equivalent to the first one, is this new normed space the dual of $X$ as well? (I think it ...
2
votes
1answer
115 views

How to show $0$ is a point of closure of weak topology, but not a limit of weakly covergent sequence in a a subset of $l^2$

(von Neumann)For each natural number $n$, let $e_n$ denote the sequence in $\mathcal {l}^2$ whose $n$th component is $1$ and other components vanish. Define$$E = \{e_n + n \cdot e_m : n,m \in \Bbb ...
2
votes
2answers
93 views

Proving the normed linear space, $V, ||a-b||$ is a metric space (Symmetry)

The following theorem is given in Metric Spaces by O'Searcoid Theorem: Suppose $V$ is a normed linear space. Then the function $d$ defined on $V \times V$ by $(a,b) \to ||a-b||$ is a metric on $V$ ...
0
votes
1answer
170 views

Differentiable function in the normed space $\,\mathcal{C}[0,1]$

Let $E=\mathcal{C}[0,1]$ with norm $\|\cdot\|_\infty$. Let $f:E\rightarrow E$ defined by $f(x)(t)=\sqrt{|x(t)|}$. For which functions $x$ is differentiable?
0
votes
1answer
39 views

$W(A)=\{x^HAx : x^Hx=1,{x\in \mathbb{C}}\}$, ${A\in \mathbb{R}}^{n\cdot n}$ How do I show that set is symmetrical set regard to real axis?

I need help to solve this task, so I would accept any suggestion: If ${A\in \mathbb{R}}^{n\cdot n}$, show that set $W(A)=\{x^HAx : x^Hx=1,\,{x\in \mathbb{C^n}}\}$, is a symmetrical set with respect ...
5
votes
1answer
112 views

How to show that $I-T$ is surjective, if $\|T\|<1$?

I got stuck on an exercise in page 258, Real analysis(4ed),H.L. Royden et al: Let $X$ be Banach space and $T \in \mathcal{L}(X,X)$ have $\|T\|<1$. Show that $I-T$ is an isomorphism. ...
3
votes
1answer
65 views

$K$ is weakly-compact $\Longleftrightarrow$ $\Pi(K)$ is weak*-compact

Let $X$ be a Banach space and $K\subset X$. $\displaystyle \Pi:X \longrightarrow X$** canonical injection $\Pi(x)(f)=f(x)$ How can we prove that: $K$ is weakly-compact $\Longleftrightarrow$ ...
1
vote
1answer
97 views

The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable

Let $X$ be a Banach space. If $B\subset X$* is a norm-separable How can we prove that: The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable. $X$*$=B(X,\mathbb{R})$ : ...
2
votes
1answer
38 views

How to put a norm on an ultrapower of normed spaces?

I'm trying to understand how to form the ultrapower of a normed space. I read how to construct the underlying vector space here: http://en.wikipedia.org/wiki/Ultraproduct But I don't see an obvious ...
3
votes
1answer
117 views

A non-separable strictly convex space with a separable pre-dual

I am asked to find an example of a non-separable strictly convex space with a separable pre-dual. So please give me some hints or some references about such problems. Thanks in advance.
4
votes
3answers
842 views

equivalent norms in Banach spaces of infinite dimension

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that $$ \forall x \in X, \|x\|_1 \leq ...
0
votes
2answers
39 views

Prove that $\lim_{n\rightarrow+\infty}\|\sup_{k>n}u_k-u_n\|=0$

Let $\{u_n\}$ be a Cauchy sequence in the space $(\mathbb R,d)$ with $d(x,y)=\|x-y\|$. Prove that $$\lim_{n\rightarrow+\infty}\|\sup_{k>n}u_k-u_n\|=0.$$ This seems to be obviously however I can not ...