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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3
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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
83 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
201 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
72 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
205 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 ...
0
votes
1answer
122 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
42 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
vote
1answer
104 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
75 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
141 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
110 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
87 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
161 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
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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
110 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
63 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
96 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
116 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.
3
votes
3answers
722 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
37 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 ...
1
vote
1answer
92 views

Is a normed topological space metrizable?

As stated in the title: If there is a norm on a topological space, then we get a metric induced by the norm. Is this true?
3
votes
2answers
581 views

Derivation of the polarization identities?

For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either $$ \langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$ or $$ \langle ...
2
votes
1answer
564 views

Weakly closed implies sequentially closed

Another problem involving the weak topology: Let $X$ be a normed space and $A \subset X$ weakly closed. Then $A$ is sequentially closed, that is: If $(x_n) \subset A$ and $x_n \xrightarrow{w}x$, then ...
2
votes
1answer
38 views

Compactness and Normed Linear spaces

If the set $S=\{ x \in X : ||x||=1 \}$ in the normed linear space $X$ is compact, how can it be shown that $X$ is finite dimensional?
1
vote
1answer
98 views

A trouble about the Simons’ inequality

I have a trouble in the proof to Simons’ inequality: About prove that: $\displaystyle \inf_{x \,\in\, C_1} \sup_{B} (x) \le \sup_{B} (\lim_{n} \sup (x_n)) \Longrightarrow \sup_{B} (\lim_{n} \sup ...
0
votes
1answer
55 views

A metric space $(\Bbb R,d)$ with $d(x,y)=||x-y||$ is complete!

I would like to receive only the hint, how to prove the statement on the heading. I understand that we have to prove that all Cauchy sequences converges in the space $\Bbb R$, e.g. ...
2
votes
1answer
45 views

A trouble about the Ekeland variational principle

I have a trouble in the proof to $EVP$ theorem: About the existence of the $\lim (\varphi(y_n))$ ? Any hints would be appreciated.
3
votes
2answers
73 views

Relations between $\|x+a\|$ and $\|x-a\|$ in a normed linear space.

1) Can it happen that $\|x+a\|=\|x-a\|=\|x\|+\|a\|$ when $a\ne0$? 2) How large can $\min(\|x+a\|,\|x-a\|)/\|x\|$ be when $\|x\|\ge \|a\|$? (For a inner-product space, the answers are no and ...
0
votes
1answer
38 views

If $K$ is $w$−compact and convex, $f\in X^\ast \implies f$ attains its maximum on $K$

Let $X$ be a real Banach space If $K\subset X$ is weakly compact and convex, then for a given $f\in X^\ast$ (dual space) we can always find $k\in K$ such that $$\displaystyle \sup_{x\in ...
1
vote
2answers
102 views

Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n ...
4
votes
1answer
95 views

Weak and strong topology on infinite dimensional spaces

Is there a simple example to show that the weak and strong topology on an infinite-dimensional space do not need to coincide? I have several ideas using differences in the weak and strong convergence ...
0
votes
1answer
210 views

Weakly bounded iff uniformly bounded in $E'$?

I have a problem: Suppose that $E$ be a normed space over $\mathbb{R}$ and $E= \{f: [0,1] \to \mathbb{R}\ \text{is continuous and such that}\ f|_{[0, \delta]}=0, \text{with}\ \delta=\delta(f)>0 ...
2
votes
1answer
401 views

Difference between convergence in norm, point-wise and uniform convergence

I know both definitions but I was wondering what are the relations between them. My question is if someone could explain intuitively the differences between these types of convergence. Specifically, ...
1
vote
3answers
177 views

Bounded linear operator in a normed space, $Tf=f(0)$ on $C[0,1]$

Let $E = C[0,1]$ with the norm $\|\cdot\|_\infty$. Define $T:E\rightarrow\mathbb{R}$ as $Tf=f(0)$. Prove that $T$ is a bounded linear operator and calculate $\|T\|$. I already tried to prove that T ...
0
votes
1answer
19 views

Decreasing sequence in a normed space

Consider by $p\geq 1$ the set $l^p=\{(x_n):x_n\in\mathbb{R},\,\,\sum |x_n|^p<\infty\}$. If defined by $x\in l^1$ $$||x||_p=\left(\sum_{n=1}^{\infty} |x_n|^p\right)^{1/p}$$ How to prove that the ...
0
votes
1answer
125 views

Existence of a special linear subspace (of the dual of a subspace)

Let $X$ be a normed space and $Z \subset  X^*$ a separable linear subspace. Then there is a separable linear subspace $Y \subset X$ such that $Z$ is isometrically isomorphic to a linear subspace of ...
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vote
1answer
53 views

the$|.|_2$ norm properties

I have come across a lemma whose proof I do not quite get. $$ $$ Lemma. Let $x_i \in \Omega^N$. Then, for any $\varphi \in C^2(\overline{\Omega})$, $$|(D^- - \frac{d}{dx})\varphi(x_i)| \le ...
4
votes
1answer
141 views

Exercise of series in a Banach Space

A vector basis of a vector space $E$ is a family $(a_{\lambda})_{\lambda\in L}$ such that any element of $E$ can be written in a unique way as a linear combination of a finite number of $a_{\lambda}$, ...
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0answers
53 views

Exercise over a strictly increasing sequence of vector subspaces of $E$ of finite dimension

Let $E$ be a real normed space of infinite dimension, and let $(E_n)$ be a strictly increasing sequence of vector subspaces of $E$ of finite dimension. Show that there exists a sequence $(x_n)$ of ...
0
votes
1answer
40 views

Equality series exercise in normed spaces

Let $\sigma$ be a bijection of $\mathbb{N}$ onto itself, and for each n, let $\sigma(n)$ be the smallest number of intervals $[a, b]$ in $\mathbb{N}$ such that the union of these intervals is ...
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0answers
54 views

Sequence in a normed space exercise

Let $(a_n)$ be an arbitrary sequence in a normed space $E$. Show that there exists a sequence $(x_n)$ of points of $E$ such that $\,\,\,\displaystyle{\lim_{n\rightarrow \infty}x_n=0}\,\,\,$ and a ...
4
votes
1answer
312 views

Does there exist such a closed subspace of normed linear space

let $(X,|| || )$ be a norm linear space. And $M$ be a closed subspace of norm linear space .does there exist a closed subspace $N$ such that $X=M \oplus N $ . I know such an subspace $N$ exist .but i ...
1
vote
1answer
202 views

Every quotient of a reflexive space is reflexive

How do you prove the following? If $\mathcal{X}$ is reflexive and $M \leq \mathcal{X} \rightarrow \mathcal{X}/M$ is reflexive There is no assumption that $\mathcal{X}$ is a Banach space.
5
votes
5answers
451 views

Fixed point Exercise on a compact set

Let $K$ a compact normed space and $f:K\rightarrow K$ so that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ have a fixed point.
3
votes
1answer
319 views

If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable

Let $X$ be a Banach space such that $X$* (Dual space of $X$) is separable How can we prove that $S_{X^\ast}$ (Unit sphere of $X$*) is also separable Any hints would be appreciated.
2
votes
2answers
397 views

Proof of uniqueness of the bounded linear transformation extended in the Bounded Linear Transformation theorem

B.L.T Theorem (from Reed/Simon): Suppose $T$ is a bounded linear transformation from a normed linear space $\langle V_1, \|\cdot\|\rangle$ to a complete normed linear space $\langle V_2, ...
2
votes
1answer
177 views

Adaptation to Banach–Mazur theorem

I'm trying to prove the following: For every normed linear space $X$, there exists a isometric ismorphism of $X$ in $C(K)$, where $K$ is a compact space. I know from Banach–Mazur theorem that ...
5
votes
1answer
59 views

Equivalent continuation of a metric

Hello fellow mathematicians, I am confronted with the following, supposedly not too difficult, problem: Let $(E,f_1)$ be a normed space and $F \subset E$ a linear subspace. Let $f_2$ be a norm on E ...