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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1answer
87 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?
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2answers
559 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
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1answer
546 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
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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?
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1answer
95 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 ...
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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
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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.
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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 ...
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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 ...
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2answers
98 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
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1answer
94 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 ...
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1answer
208 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
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1answer
395 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, ...
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3answers
176 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 ...
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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 ...
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1answer
124 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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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 ...
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1answer
138 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 ...
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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 ...
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1answer
305 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 ...
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1answer
201 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.
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5answers
440 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.
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1answer
314 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.
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2answers
381 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
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1answer
172 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
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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 ...
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3answers
74 views

One-point compactification of a linear space

Is it possible to take the one-point compactification of a linear space and get again a linear space? Like if $X$ is a normed space add $*$ to it and define $|x-*| = 1$?
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1answer
848 views

$C([0, 1])$ is not complete space with respect to norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Consider $C([0, 1])$, the linear space of continuous complex-valued functions on the interval $[0, 1]$, with the norm $\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx$. I have to ...
2
votes
1answer
84 views

Convergence of coordinates to zero

Consider a normed finite-dimensional vector space $V$ with some norm $|| .||$ Say a sequence of vectors in this vector space $v_m \rightarrow 0$ where $0$ is the zero vector. Let ...
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0answers
29 views

Absolute summable sequence in normed space [duplicate]

I have studied that for a sequence of real numbers absolute summability implies summability. What can we say about the sequences $\{x_k\}$ in a normed space . If it is not true in general could ...
2
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1answer
87 views

Gateaux and Fréchet differentials in $\ell^1$

I am in trouble trying to solve the following: Let $X = \ell^1$ with the canonical norm and let $f \colon \ell^1 \ni x\mapsto \Vert x \Vert$. Then $f$ is Gateaux differentiable at a point $x = ...
2
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1answer
91 views

Projection on unit vector lipschitz condition

if i have a vector $ v \in C^{n}$, where we do not want to look at vectors with norm $||v||\le 1$ with an arbitrary norm of this space and I am asked whether the map $ f(x)=\frac{x}{||x||}$ satisfies ...
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1answer
64 views

Why this space is not a complete space with this norm

Show that the space $C_0(\mathbb{R})$ of all the real continuous functions $f:\mathbb{R} \to \mathbb{R}$ with compact support is not a complete space with the norm $||f||= \sup_{t∈ \mathbb{R}}|f(t)|$. ...
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2answers
135 views

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} ...
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2answers
507 views

Proving that Euclidean space having the infinity metric is a complete metric space (stuck)

I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space. I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
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2answers
73 views

Exercise of interior of a closed ball

Let $E$ a normed spaces and $a\in E$. How to prove that $$[\overline{B}(a, r)]^{\circ}\subseteq B(a, r)$$ where $\overline{B}(a, r)=\{x\in E: \|x-a\|\leq r\}$ and $B(a, r)=\{x\in E: \|x-a\|< r\}$ ...
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2answers
149 views

Two terms that I want to understand: weakest topology and jointly continuous (in the following context).

I was reading an article online, please help me to understand the following lines (in bold letters). - Topological structure: If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
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0answers
100 views

Does every Banach space admit a continous injection to a non-closed subspace of another Banach space?

Let $(V, ||\,||)$ be a Banach space. I want to produce a non-complete norm $||\,||'$ on it such that $||v||' \leq ||v||$ for all $v$ in $V$. Given a continuous injection $\varphi\colon V \to W$ with a ...
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2answers
142 views

Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?

I hope that my question in the title is well formulated. I am a little bit confused with the next exercise from a book: Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ...
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1answer
120 views

Why can I choose elements of $X$ (and $h \in L^2(0,T)$) in this way? (Dual spaces, norms, Bochner spaces)

This is from the book Vector Measures by Diestel and Uhl, page 98: Let $X$ be a Banach space. Let $\epsilon > 0$ and suppose first that $g = \sum_{i=1}^\infty x_i^* \chi_{E_i}$ where $x_i^* \in ...
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4answers
211 views

Show that $c$ is closed in $l^{\infty}$

Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$ $$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
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2answers
233 views

How to expand equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have a regression $Y=X\beta+\epsilon$. I want to solve (meaning expand), $$\displaystyle\|Y-X\beta \|_{2}^2$$ Should I do: 1) ...
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1answer
54 views

Function Spaces

What is exactly the difference between $L^2$ space and ${\ell}^1$ space? I believe that one of them is the space of square of square integrable functions. Does it have to do with one is for series ...
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1answer
54 views

Existance of function of norm one on normed spaces

On a past exam in a course on functional analysis the following problem is given: Let $X,Y$ be normed spaces and let $x\in X$ be nonzero. Show that there exist some $f\in X^\ast$ s.t. $f(x) = \|x\|$ ...
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1answer
71 views

Good source for Triebel-Lizorkin spaces?

I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have $L^p$ and $W^{k,p}$. The next function spaces I thought I'd ...
2
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1answer
133 views

Inequality for norms

Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$ $$ \|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)} $$ Thsnk you.
2
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2answers
228 views

Equivalent norms and isometries

Let $X$ be a vector space, $||\cdot||_1$ and $||\cdot||_2$ two equivalent norms on $X$. Under what further assumptions can we prove there is an isometry between $(X,||\cdot||_1)$ and ...
9
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2answers
673 views

$C[0,1]$ is complete w.r.t. which norm(s)

$C[0,1]$ is complete w.r.t. which norm(s) $\displaystyle\|f\|_\infty=\sup_{t\in[0,1]}|f(t)|$ $\displaystyle\|f\|_1=\int_0^1|f(t)| \, dt$ $\displaystyle\|f\|_\infty^{0,1}=\|f\|_\infty+|f(0)|+|f(1)|$ ...