1
vote
1answer
41 views

How to check if a given piecewise defined function on $\mathbb R^2$ is a norm?

I want to check if the function $\parallel (x,y)\parallel = \left\{ \begin{array}{cc} \sqrt{x^2+y^2} & \mbox{if } xy \geq 0 \\ \max\{\vert x\vert, \vert y\vert\} & \mbox{if } xy < 0 ...
1
vote
1answer
31 views

Show there exists a Cauchy subsequence

Let $X$ be a separable reflexive real Banach space and $\{\psi_n\}$ be a dense sequence in $$\{\psi\in X' : ||\psi||_{X'} \leq 1\}.$$ Consider in $X$ the scalar product defined by $$(x | y)_0 = ...
1
vote
1answer
31 views

Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
2
votes
1answer
40 views

Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1 $)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1 $) if they induce equivalent metrics. At first I thought this ...
0
votes
1answer
32 views

Lengths of curves - Arc length

If $\gamma\colon [a,b] \rightarrow \mathbb{R}^d$ is a differentiable curve we usually define $$ \text{length}(\gamma) := \int_a^b \|\dot{\gamma}(t)\|_2\, \text{d}t,$$ where $\|\cdot\|_2$ is the ...
0
votes
1answer
20 views

Open set generated by two equivalent norm

Suppose X is an vector space and $||.||_a$ and $||.||_b$ are two norm on it, how i can proof that ''This two norms are equivalent iff they generate same open sets."? P.S.: Sense of question made by ...
1
vote
1answer
56 views

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$, then $f = Cg$ for some non-negative constant $C$. First assume $||f ||_{L^p} +||g||_{L^p} = 1$, ...
1
vote
1answer
25 views

Norm of the functional $f(x)=\int_0^2 x(t).(t^2-1)dt$ in the space $X=L_1(0,2)$.

I am trying to calculate the norm of the functional $f(x)=\int_0^2 x(t).(t^2-1)dt$ in the space $X=L_1(0,2)$. What we have is: $||f(x)||\le ||x||_1.||t^2-1||_{\infty}$ by extended Hölder's ...
1
vote
2answers
56 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
5
votes
1answer
71 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
1
vote
1answer
107 views

Fredholm operator norm

I have seen here, that the operator norm of a Fredholm operator $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$ is not equal to the $L^2$ norm of the Kernel. ...
0
votes
0answers
24 views

For what range of $a\in\Bbb R$ is $f(x)=ae^{-ax}$ in the unit ball with respect to the uniform metric on $[0,1]$

For what range of $a\in\Bbb R$ is $f(x)=ae^{-ax}$ in the unit ball with respect to the uniform metric on $[0,1]$? I.e when is $\|f\|_\infty\lt 1$ So far I see that for $a\ge 0$, ...
1
vote
0answers
25 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
1
vote
1answer
35 views

Questions over a specific case of the Muntz-Szasz theorem proof

On page 157 of this site: http://arxiv.org/pdf/0710.3570.pdf the author is proving a specific case of one direction of the Muntz-Szasz theorem. I do not understand the following 3 claims: 1) For ...
0
votes
0answers
42 views

limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
1
vote
1answer
43 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
0
votes
2answers
31 views

Ratio of $\|\cdot\|$ and $\|\cdot\|_{\infty}$ on $\mathbb{R}^2$

I have the following question from an old examination paper in Real Analysis: On $\mathbb{R}^2$, $\|\cdot\|_{\infty}$ is defined by ...
0
votes
1answer
42 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
2
votes
1answer
56 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
0
votes
3answers
34 views

Is this some kind of triangle inquality?

I stumbled upon the following inequality: $$\Vert x+hz-(x+y)-(p-(x+y))\Vert_2 \geq \Vert p-(x+y)\Vert_2-\Vert x+hz-(x+y)\Vert_2$$ where $p,x,y,z \in \mathbb{R}^n$. My question is: Is this some kind ...
1
vote
1answer
26 views

Check continuity of linear functionals and find norms

1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$ where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm 2) $\ell^\infty \owns (x_n) \mapsto ...
1
vote
2answers
52 views

Equivalence of norms proof

This question is from a set of optional, much harder problems from my first year analysis course, but the subject material is norms on $\mathbb R^K$. (c) Show that there exists a constant $C > 0$ ...
1
vote
2answers
35 views

Positivity of a function in $\mathbb{R}^{n}$

We place ourself in $\mathbb{R}^{n}$. We consider a given increasing function $$ g : \begin{aligned} &\mathbb{R}^{+} \to \mathbb{R} \\ &x \;\;\,\mapsto g(x) \end{aligned}$$ Finally, we ...
0
votes
0answers
58 views

Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
2
votes
2answers
109 views

How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$?

Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks!
0
votes
1answer
46 views

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm?

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm ? I know that the algebra of polynomials is dense in algebra of continuous functions, wrt to sup norm, And I know that if $f ...
2
votes
2answers
65 views

Comparable norms on the space of polynomials?

Are the norms: $$\|P\|_1=\int^1_0\|P(t) \| dt\mbox{ and }\|P\|_2=\sup_{0\le t\le1} |P(t)|.t$$ comparable on the vector space $X=\mathcal{P}[0,1]$, i.e. all polynomials on $[0,1]$? Here, i try to ...
0
votes
1answer
40 views

Prove that the operator norm is a norm

Exercise: Prove that the operator norm of the set $S$ of all linear operators $L:R^n\to R^m$ defines a norm on $S$ Definition of norm: A positive function $\| .\|$ on a real vector space $V$ is a ...
0
votes
0answers
33 views

Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
0
votes
1answer
62 views

proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
1
vote
1answer
51 views

When sup norm,i.e. $\| x|| = \sup|x(t)|$ for $\forall t\in T $in C[0,1] for $T \subsetneq [0,1]$

When sup norm, i.e. $\| x|| =\sup|x(t)|$ for $\forall t\in T $in C[0,1] $T$ is such that $T \subsetneq [0,1]$. What condition should be applied to $T$ to make $\| x||$ a norm. I cannot show for ...
0
votes
1answer
39 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
0
votes
1answer
21 views

What happens to $l_1$ if i change coordinate system.

Let $x =(x_1,\ldots,x_n) \in \mathbb{R}^n$ and also $x=\sum_{i=1}^m t_i u_i,$ where $t_i \in \mathbb{R}$ and $u_i \in \mathbb{R}^n.$ Is it true that $||x||_1 \geq \sum_{i=1}^m |t_i|$ ?
3
votes
2answers
99 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
0
votes
2answers
47 views

Is convergence in the norm equivalent to convergence of norms?

If $\| \cdot \|$ is a norm on some space. Does the equivalence go both ways? $$\| f_n-f \| \to 0 \Longleftrightarrow \| f_n\| \to \| f\| $$ The $\implies$ direction is obvious since $\| f_n-f \| ...
1
vote
1answer
53 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
0
votes
3answers
60 views

closed and bounded but not compact set of real-valued bounded functions

I'm trying out a problem I was given and this is the statement: Prove, or disprove, that every bounded and closed subset of the set of real-valued and bounded functions on [0,1] equipped with the sup ...
3
votes
1answer
116 views

Prove that the given distance function is a norm

Let the vector space $X=K^3$. For $x=(\alpha_{1}, \alpha_{2}, \alpha_{3}) \in X$, we define $||x||= [(|\alpha_{1}|^2+|\alpha_{2}|^3)^\frac{3}{2} + |\alpha_{3}|^3]^\frac{1}{3}$ Proof that $||·||$ is ...
0
votes
2answers
94 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
2
votes
1answer
31 views

Proving that $||f(x)||$ is Riemann integrable

Suppose $f=(f_1,...,f_m)$ be a vector valued function which is continuous on $[a,b]$. How can I show that $||f(x)||$ is also Riemann integrable? Any answers will be much appreciated. Thanks
3
votes
0answers
40 views

Norms on $\mathbb{Q}$

So with respect to the metric $d(x,y)=|x-y|$ induced by the standard absolute value, the real numbers can be constructed as a completion of $\mathbb{Q}$. With respect to the metric $d_p(x,y)=|x-y|_p$ ...
1
vote
1answer
57 views

continuity of $L^p$ norms with respect to $p$

Let $0<p_0<p<p<p_1\leq \infty$. Then I have proved $L^{p_0}(\mu)\cap L^{p_1}(\mu)\subseteq L^{p}(\mu)$. In particular, when $p_0=1$, $p_1=\infty$, I have proved further ...
2
votes
1answer
57 views

Triangle inequality in product space of normed spaces

Let $(X,||.||_X)$ and $(Y,||.||_Y)$ be normed spaces, then $||(x,y)||:=(||x||_X^p+||y||_Y^p)^{\frac{1}{p}}$ is a norm on $X \times Y$. This is absolutely clear to me, but I have troubles to verify ...
0
votes
1answer
39 views

Norm of a vector is determined by evaluation of linear functionals on it: can this be proved without the Hahn-Banach theorem?

Let $V$ be a normed vector space over the field of real numbers, $\mathbb R$, and let $x_0 \in V$ be fixed. I know how to prove $$\|x_0\| = \sup_{f \in V^*, \|f\| = 1} |f(x_0)|$$ using the Hahn-Banach ...
1
vote
1answer
48 views

Norm of a linear mapping, please check if what I have done is right

please check if what I have done is right. $C[0,1]=$ continuous functions in $[0,1]$ considering $\|g\|=\max_t|g(t)|$ $$X=\langle t^2,1 \rangle $$ the subspace of $C[0,1]$ generated by $t^2$ and $1$ ...
2
votes
1answer
110 views

Strange mistake in this proof about normed separable vector spaces.

I am supposed to prove that an infinite-dimensionale separable normed space $X$ constains a countable subset $Y$ that is linearly independent and dense. There are three hints given: First, prove ...
1
vote
1answer
106 views

does the max function holds the triangle inequality?

I need to prove if the following is a norm: $$||f||:=\max_{-1<x<0}|f| + \max_{0<x<1}|f|$$ when $f$ is a continuous on $[-1,1]$. The only problem I have is with showing it holds the ...
1
vote
1answer
65 views

Space on convergent sequences - is it an inner product space

I'm trying to prove some properties of sequence spaces. I already know that the space $l^{\infty}$ of all bounded sequences isn't an inner product space, isn't separable but it is complete with $sup$ ...
3
votes
0answers
134 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
1
vote
1answer
60 views

Example of two norms and ONE linear operator that is bounded and unbounded in a norm.

I am looking for an example of a linear operator that is bounded as well as unbounded depending on which norm you take. Since I do not have much experience with Functional Analysis, I do not know many ...