0
votes
1answer
33 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 ...
1
vote
1answer
43 views

Inequality with a norm

I need help with the following: Let $A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right)$, with $a\in\mathbb{R}$, $b\in(l^{1})^{*}$, $c\in l^{1}$, and $d\in L(l^{1},l^{1})$. Let $h\in ...
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. ...
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 ...
4
votes
2answers
51 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.
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
1answer
43 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 ...
1
vote
0answers
50 views

proving uniform convergence for a uniformly continuous function

I've been stuck on this question (2a) for about a day now! Question and havn't really gotten anywhere, I think that the series is that integral of a step function that approximates f ie: $\Gamma_n ...
1
vote
1answer
37 views

Check continuity and find norm of linear functional

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, ...
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
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 ...
1
vote
1answer
38 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
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 ...
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 ...
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: ...
3
votes
1answer
117 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 $||ยท||$ is ...
0
votes
2answers
102 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 ...
0
votes
1answer
56 views

Norm of a bounded operator

Let $\phi \in C[0,1]$ and $T_{\phi}:C[0,1] \rightarrow \mathbb{R}$ be: $T_{\phi}f = \int_0^1 f(x)\phi(x)dx$ prove that $T_{\phi}$ is a continuous, linear functional and that $||{T_{\phi}}|| = ...
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 ...
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
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 ...
0
votes
0answers
15 views

How to prove $||A||_2\leq ||A||_F \leq \sqrt{n}||A||_2$ [duplicate]

$A$ is a square matrix with dimension $n$ and $||A||_F$ is Frobenius norm.
2
votes
1answer
251 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
1
vote
1answer
73 views

How an $\ell_1$ Inequality Implies Equality

Suppose that for scalar $\epsilon$ we know that $\vert \epsilon \vert$ is small enough such that the sign pattern on $\mathbf{x}\in\mathbb{R}^n$ is equal to that on $\mathbf{x} + \epsilon \mathbf{h}$, ...
2
votes
2answers
94 views

What is the definition of the norm

Let $x$ and $y$ be in $ \mathbb{R}^{n}$. I know from the definition of norm that $\|x\|=\sqrt{\sum_{1}^{n}x_{i}^{2}}$. Can anyone tell me what will be the norm of $\|x-y\|$? Is it ...
3
votes
1answer
102 views

Show if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is linear and invertible

I want to show that if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is a linear and invertible function. First I need to show if $x\neq0$ then $\|f(x)\|>0$. Since $f$ is ...
0
votes
1answer
103 views

Showing a norm preserving isomorphism of vector spaces

Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
0
votes
1answer
182 views

Showing triangle inequality for a norm

I want to determine whether the following is a norm or not: \begin{equation} ...
4
votes
0answers
106 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 ...
0
votes
1answer
96 views

Sobolev spaces doubt

Can somebody help me with this doubt? Let $\Omega$ an open set and $A$ be any finite subset of points of $\Omega.$ Is it true the following inequality? $\vert v(a) \vert \leq C \| v \|_p ...
0
votes
3answers
535 views

How to find the norm of this bounded linear functional?

Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
0
votes
1answer
175 views

Analysis.. Norm on C([a,b])

Let $w:[a,b]\rightarrow \mathbb{R}$ with $ w(x)\geq c>0 $ for some $c \in \mathbb{R}$ and all $x \in [a,b]$. Prove that $$\lVert f\rVert_w \ = \ \displaystyle\int^b_a \lvert f(t)\rvert w(t)\ ...
0
votes
1answer
80 views

Analysis.. Convergence of sequence

I really struggle with understanding convergence and have the following questions.. Determine whether the following sequences converge and if so, give the limit: $x_n = ...
2
votes
2answers
76 views

Norms Abstract Analysis

I have a question relating to norms and have been giving functions and need to state whether they are norms or not... which of the following are norms on $\mathbb{R}^2$? Give reasons for your ...
2
votes
0answers
155 views

Step functions dense in Integrable functions with respect to $L_2$

Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
3
votes
0answers
74 views

Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show that ...
2
votes
1answer
392 views

Does the limit of a convergent sequence depend on the norm?

Let $X$ be a vector space, and $\|\cdot\|_1$ and $\|\cdot\|_2$ two different (non-equivalent) Norms on $X.$ Let $(x_n)\subset X$ be a sequence and $x\in X$ such that $\lim_{n\to\infty}\|x_n-x\|_1=0.$ ...
1
vote
2answers
80 views

what does it mean to say a space is norm separable?

I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
3
votes
1answer
159 views

Norm of operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
1
vote
2answers
259 views

equivalence of two definitions of norm equivalence: “$|\cdot|_1=|\cdot|_2^\alpha$” vs. “being a Cauchy sequence is the same for both norms”

Let $|\cdot|_1$ and $|\cdot|_2$ be two norms on a field $\mathbb F$. We call the two norms equivalent if every Cauchy-sequence with respect to $|\cdot|_1$ is also a Cauchy-sequence with respect to ...
1
vote
3answers
396 views

supremum norm and submultiplicativity

If $f$, $g \in C(S)$ where $S$ is a compact set in $\mathbb{R}^n$ then it is true that $$\lVert fg \rVert \leq \lVert f \rVert \lVert g \rVert$$ where the norm is the usual supremum norm. Why is this ...
1
vote
1answer
82 views

Analysis simple question

Let $S= \{(x_1,\ldots, x_n)\in \mathbb{R}^n$; $|x_1|^p+\ldots+|x_n|^p=1\}$, where $p>1$ is real(and fixed), consider a fixed $y\in\mathbb{R}^n$ and $T:\mathbb{R}^n\rightarrow\mathbb{R}$ such that ...
2
votes
1answer
122 views

Limit inferior taken on the norm of a sequence

Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$. Why is it that from the inequality $$ |f(x_n)| \leq \|f\| ...
1
vote
2answers
102 views

Equivalent norms on $\mathbb{R}^2$

For $(\mathbb{R}^2,\|\cdot\|_2)$ and $(\mathbb{R}^2,\|\cdot\|_\infty)$ and any $x \in B((0,0),1,\|\cdot\|_2)$ how would you find a $\delta_x$ such that $B(x,\delta_x,\|\cdot\|_\infty) \subset ...
2
votes
2answers
124 views

Proving that $\|x\|_2 \geq \|x\|_1$?

How would you prove that $\|x\|_2 \geq \|x\|_1$, or in other words that $$\sqrt{\int_0^1|f(x)|^2 dx} \geq \int_0^1|f(x)|dx \quad?$$
3
votes
1answer
381 views

Minimize distance between 2 functions

Just so you know, this is a homework question, and I basically need help with the steps to solve this problem. I understand what it's asking; however, my attempts haven't worked out, and I'm probably ...
2
votes
1answer
80 views

P-adic “Norm” and scalability criterion

I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ? What I mean ...
3
votes
1answer
275 views

Norm with special conditions

Let $N$ be a norm on $\mathbb R^2$, such that $N ( \mathbb Z^2) \subset \mathbb N $, where $\mathbb Z^2 =\{ (a,b)\mid a\mbox{ and }b \mbox{ are integers}\}$. Help me to prove that for $u$, $v$ fixed ...
3
votes
1answer
815 views

Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$

I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...