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.
25
votes
0answers
430 views
Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$
Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on ...
5
votes
0answers
187 views
Are Arzelà–Ascoli theorems results of similar theorems on normed spaces, metric spaces or other spaces?
From Wikipedia, two generalizations of the Arzelà–Ascoli theorem are
Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$,
the set of real-valued continuous functions on X, is ...
4
votes
0answers
75 views
Is the result still true if we drop completeness?
I know how to prove the following exercise ( from Folland) :
If $X$, $Y$ are Banach spaces. $T:X\rightarrow Y$ is a linear map such that $f\circ T\in\operatorname{dual}(X)$ whenever $f\in ...
4
votes
0answers
143 views
Metric on the unit cube
Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y,y\neq -x$ define $d(x,y)$ to be the arc length of the path $$Y\cap \{\lambda ...
3
votes
0answers
60 views
Find Lipshitz constant
Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution.
Find Lipshitz constaants of the following functions:
$$
f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i
...
3
votes
0answers
50 views
What are norms used for?
These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
3
votes
0answers
38 views
Do these inequalities imply these inequalities? (Norms and squaring)
Suppose we have the inequalities involving norms
$$\lVert f \rVert_{X}^2 \leq C_1(\lVert f \rVert_{Y}^2 + \lVert f \rVert_{Z}^2)$$
and
$$\lVert f \rVert_{X}^2 \geq C_2(\lVert f \rVert_{Y}^2 + \lVert f ...
3
votes
0answers
75 views
Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?
Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then
$f\equiv 0 \rightarrow \rho(f) = 0$
when $|a| \neq 0$, ...
3
votes
0answers
92 views
Do the $p$-norms in $\mathbb R^2$ have a nice geometric intuition behind them?
For $p\geq 1,$ the $p$-norm of a vector $(x,y)\in\Bbb R^2$ is the number $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}.$
I learned this definition some time ago, but I never really understood it. Is there a ...
2
votes
0answers
61 views
Which are nontrivial examples of analytical functions on Frechet spaces?
Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
2
votes
0answers
22 views
Are these two definitions for dual norm equivalent
Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is,
$$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$
The second is,
$$
\sup\limits_x ...
2
votes
0answers
29 views
Finding an orthornormal basis given a bilinear form
Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
2
votes
0answers
100 views
$\ell^0$ and $\ell^{\infty}$ norms
Let $x \in S^{n-1}$ and such that its coordinates $|x_1|\geq \cdots \geq |x_n|$.
Under which condition on $\|x\|_0$ the following inequality is true that $$\|x\|_{\infty}\leq \frac{1}{\sqrt ...
2
votes
0answers
121 views
Continuous, Bounded Normed Linear Spaces
For $f$ in $C[a, b]$, define $||f||_1 = \int_a^b |f|$.
Show that this is a norm on $C[a, b]$.
Also, show that there is no number $c\geq0$ for which $||f||_{\max}\leq c||f||_1$ for all $f$ in ...
2
votes
0answers
100 views
Is it a Banach space? If so what is its dual?
Let $(E_n)$ be a sequence of Banach spaces and $(w_n)$ be a sequence of positive real numbers. For $1\leq p <\infty$ define $\bigoplus\limits_P E_n:=\{(x_n):x_n\in E_n,\sum\limits_n\lVert ...
2
votes
0answers
165 views
Question about proof of Stone-Weierstrass
I would like to know if I understand the details in the proof of Stone-Weierstrass (in $\mathbb R$) so I'd like to post it here in my own words. Can you please check it and tell me if it's correct? ...
2
votes
0answers
70 views
Consequence of metrizability proof - disregard, the question is an error
In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
2
votes
0answers
107 views
norms on a vector space - is there a quicker way to approach this problem?
I'm looking at the following problem, which I have done but wonder whether there isn't a faster way of doing it. (It's a past exam question which is supposed to take 7.5 minutes, but I only managed to ...
2
votes
0answers
57 views
Embedding tree metric isometrically into $\ell_\infty$
I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
2
votes
0answers
106 views
Different topologies on a normed/inner product space
Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm.
Define:
$A$:={topologies that can make the norm continuous},
$B$:={topologies that can make the ...
2
votes
0answers
145 views
Proof of equivalence in strictly convex spaces
I'm trying to understand a proof and from my current understanding there is one thing not clear/wrong. Maybe you can help:
Let $(X, \lVert\cdot\rVert)$ a normed space and $B(0,1)$ the closed unit ...
1
vote
0answers
61 views
Define metric on set and products
Let $X$ be set. My question is: if adding point $\ast$ to $X$ to get set $X \cup \{\ast\}$ then on countable product $\prod_{n \in \mathbb N_+} X \cup \{\ast\}$ I found it possible to define metric. ...
1
vote
0answers
56 views
Normed space Analysis
Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete.
I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
1
vote
0answers
46 views
Determining Similarity of Unit Vectors
I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$.
The piecewise continuity requirement ...
1
vote
0answers
51 views
Finite Dimensional TVS
Let $E, F$ topological vector spaces, $E$ normable and $T: E \longrightarrow F$ linear, compact and surjective. Show that $\mbox{dim}(F)< \infty$.
1
vote
0answers
56 views
Normed vector spaces inequalities: proving by contradiction
Often when there is some inequality that we want to prove in a normed space, the proof goes something like "Assume there's a sequence $f_n$ with $\lVert f_n \rVert = 1$..."
Would someone give me a ...
1
vote
0answers
96 views
Separation in infinite dimensional normed space
I would like to construct some counterexamples:
$E$ is an infinite dimensional normed space. Let $C, D$ be nonempty convex subsets in $E$ such that
$$
C\cap D=\emptyset.
$$
There is no vector $f\in ...
1
vote
0answers
55 views
some facts about $L_p$ space
I was wondering relations of $L_p$ spaces..
Let $E$ be a measurable set. If $E$ is of finite measure, then $L_p(E) \subset L_q(E)$, $1 \le p \le q \le \infty$.
However, does it still hold if $E$ is ...
1
vote
0answers
132 views
equivalence of norms in an open set
I would like to prove that two given norms in the space of smooth functions are equivalent in an open set, is it enough to show that they are equivalent for any compactly contained open set? why?
...
1
vote
0answers
478 views
Proof that every finite dimensional normed vector space is complete
Can you read my proof and tell me if it's correct? Thanks.
Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm ...
1
vote
0answers
65 views
Is there chance to form a frame (Riesz basis)?
Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions
$$
f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right)
$$
One can show that ...
1
vote
0answers
44 views
Dense property of $C^k_0(\Omega)$
When $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, $C^k_0(\bar{\Omega})$ is dense in $W^{k,p}(\Omega)$(Sobolev space, k-differentiable and each term estimated by p-norm).
I am wondering if it holds ...
1
vote
0answers
48 views
Disjointness of stars in a simplicial complex in $\ell_2$
Definitions
Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
1
vote
0answers
249 views
Convergence of $L^p$ norms
Given a measure space $X$ with its measure $\mu$, it can be shown (I'll provide a proof if asked for) that
$\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if } \exists p_0:\forall q \geq p_0, ...
1
vote
0answers
127 views
Neumann series in an incomplete normed algebra
Let $\mathcal{A} \equiv (A, \|\cdot\|_A)$ be a unital (associative) normed algebra over the real or complex field, and assume that $\mathcal{A}$ is not complete. Provided $\mathcal{B}_\mathcal{A}$ is ...
1
vote
0answers
43 views
What are these real-valued functions called?
Let $V\;$ be some normed $\mathbf{R}$-vector space, and $n > 0$ an integer.
First,
What is the name of the the class of mappings $\prod_i f_i(x_i):V\;^n\to\mathbf{R}$ for some ...
1
vote
0answers
451 views
Centroid under the Chebyshev distance
I want to find the centroid (point which minimizes the sum of distances) of a set of points in the 2-dimensional plane using the Chebyshev distance ($\textbf{L}_\infty$ norm). I think the answer is ...
0
votes
0answers
46 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 ...
0
votes
0answers
33 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 Hilbert space. Let $\epsilon > 0$ and suppose first that $g = \sum_{i=1}^\infty x_i^* \chi_{E_i}$ where $x_i^* ...
0
votes
0answers
16 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 ...
0
votes
0answers
48 views
a question on complete metrizable spaces
There is a claim:
Let $Y$ be a complete metrizable space. If $Y$ is bounded, i.e., $d(Y) < \infty$, then $ \exists C(X,Y)$ is a complete metrizable space.
Why here $Y$ need be bounded?
...
0
votes
0answers
24 views
Prove that $\parallel f\parallel_w=\int_a^b\mid f(t)\mid w(t)dt$ is a norm on $C([a,b])$.
Let $w:[a,b]\longrightarrow\mathbb{R}$ with $w(x)\geq c>0$ for some $c \in \mathbb{R}^+$ and all $x \in [a,b]$.
Prove that $$\parallel f\parallel_w=\int_a^b\mid f(t)\mid w(t)dt$$ is a norm on ...
0
votes
0answers
43 views
For $(x_1,x_2)$, is $G_1(x)=|x_1|$ a norm in $\mathbb{R}^2$?
For $(x_1,x_2)$, is $G_1(x)=|x_1|$ a norm in $\mathbb{R}^2$?
To prove that a function $p$ is a norm we need to prove the following:
$p(av) = |a|p(v)$
$p(u + v) \leq p(u) + p(v)$
$p(v)\ge0$, and if ...
0
votes
0answers
49 views
How to establish the scalar-multiple and the triangle inequalities as properties of the norm in the completion?
For a normed space $(X, ||.||)$, we can find a Banach space $\hat{X}$ which has a dense suspace $W$ that is isometric with $X$; and this space $\hat{X}$ is unique except for isometries.
Now while ...
0
votes
0answers
74 views
Bound for the infinity norm of a vector
Let $x$ be a vector in $R^n$ such that its support $\operatorname{supp}(x)=\frac{n}{2}$. Show that $\|x\|_{\infty}\leq \sqrt{\frac{2}{n}}$.
Thank you.
0
votes
0answers
131 views
When the linear operator is continuous.
Could I have a hint please on how to prove the following proposition:
Let $X$ and $Y$ be two normed space and $T$ be a linear
operator from $X$ into $Y$. The operator $T$ is continuous
if the ...
0
votes
0answers
151 views
Unique continuous extension of $\tilde{L}: W \to Z$
Can you read my proof and tell me if it's correct? Thank you!
Let $W$ be a dense subset of a normed vector space $V$ and let $\tilde{L}: W \to Z$ be a bounded linear operator into a Banach space. ...
0
votes
0answers
131 views
$C_c(X)$ dense in $L^p$
In class we proved that $C_c(X)$ is dense in $L^p$ where $X$ is a locally compact, $\sigma$-compact Hausdorff space either equipped with a Radon measure or equipped with a locally finite measure ...
0
votes
0answers
73 views
distance of an affine subspace to a polytope
I wonder how to prove the following statement.
Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
0
votes
0answers
45 views
existance of the interpolation space
Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following:
Is there exists space $Z\subset Y$, such that ...
