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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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
97 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
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 ...
0
votes
1answer
206 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
392 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
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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 ...
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
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 ...
1
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
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 ...
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
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 ...
1
vote
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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votes
5answers
439 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
312 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
375 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
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
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 ...
3
votes
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 $|x-*| = 1$?
2
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1answer
844 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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vote
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
votes
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
votes
1answer
90 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 ...
1
vote
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 $||f||= \sup_{t∈ \mathbb{R}}|f(t)|$. ...
1
vote
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} ...
1
vote
2answers
505 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 ${\mathbb{R}}^k$ ...
0
votes
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 $B(a, r)=\{x\in E: \|x-a\|< r\}$ ...
8
votes
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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vote
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 ...
4
votes
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 ...
1
vote
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 ...
3
votes
4answers
209 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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vote
2answers
232 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) ...
1
vote
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 ...
1
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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
69 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
votes
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
votes
2answers
223 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
votes
2answers
669 views

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

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

Where did I go wrong in showing $\|x\|_1>\sqrt n\|x\|_2$

To show $\|x\|_1>\sqrt n\|x\|_2$ where the norm $\|\cdot||_1,\|\cdot\|_2$ are defined over $\mathbb R^n$ Let $x=(x_1,x_2,\ldots,x_n),y=(1,1,\ldots,1)\in\mathbb R^n$ Using Hölder's Inequality ...
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1answer
47 views

How to verify whether $(C_{00},\|\cdot\|_p)$ is complete

How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le ...
1
vote
1answer
203 views

Relation between Normed space and inner product

Below is what i have proved: If $V$ is a normed vector space over $\mathbb{R}$ satisfies parallelogram equality, then there exists an inner product $\langle \bullet,\bullet\rangle$ such that ...
3
votes
3answers
442 views

difference between normed linear space and inner product space

I've seen that the definitions of normed linear space and inner product space for a complex vector space $V$ are very close to each other except for the fact that one is defined on $V$ and the other ...
1
vote
1answer
56 views

Unique continuos linear function given a continuous function from a dense space in X to Y (Y is a Banach Space).

Let $X$ be a normed space, let $Y$ be a Banach Space, let $D\subseteq X$ be a dense linear subspace of $X$ and let $L:D\rightarrow Y$ be a continuous linear function. Then there is a unique continuous ...
5
votes
2answers
81 views

Is this a correct question?

This is an exam question in functional analysis which for me doesn't make sense the way it is written. I am asking you if you agree with me on the modifications that needs to be done in the question. ...
2
votes
1answer
189 views

Space of finite dimensional subspaces is separable

In Bernard Maurey's paper "A Note on Gowers' Dichotomy Theorem" at the top of the 7th page, the following fact is stated that I'm not able to prove: Let $X$ an infinite dimensional separable normed ...