Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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does $\|A\| > \|B\|$, $A$ and $B$ matrices, imply that $\|Ax\| >\|Bx\|$ for all $x$ in some vector space?

I'm wondering if I can prove the first inequality in a question that I am working on, does that make the second inequality automatic? Thanks,
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241 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
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43 views

Subset of the sequence space that's closed and bounded but not compact

Consider the sequences space $l^1 = \{a = (a_n)_{n \in \mathbb{N}_0} \subset \mathbb{C}, \sum_{n = 0}^\infty|a_n|< \infty\}$ with the norm $||a||_1 = \sum_{n = 0}^\infty|a_n|$. I want to show that ...
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190 views

Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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36 views

When does the $L_1$ convergence imply almost everywhere convergence?

I know that $L_1$ convergence implies existence of an almost everywhere converging subsequence. But I was wondering, can you tell me some extra conditions on functions that make $L_1$ convergence ...
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118 views

Supremum vs. Maximum in the definition of the Lp norm [duplicate]

The $L_p$ norm $||A||_p$ is defined as $$\sup_{x \neq 0} \frac{||Ax||_p}{||x||_p} = \max_{||x||_p = 1} ||Ax||_p \tag{1}$$ I'm not quite getting why the LHS uses $\sup$ but the RHS uses $\max$. I ...
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93 views

find linear functional norm

$C[-1,1] $ above $$ f(x)=\int_{-1}^{0}x(t)dt-\int_{0}^{1}x(t)dt$$ What is norm of the f linear fucntional? I tried to solve using definition of norm but I couldn't find result. Please can you give ...
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28 views

An infinite number of solutions available to the sparse representation problem

I would like to analyze the following problem (different cases leads to which solutions to the problem and such): $$||y-Dx||_2 \leq \epsilon$$ (an overcomplete dictionary matrix $D \in \mathbb{R}^{n ...
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30 views

Is it true that $|a_i| \leq |b_i|$ implies $ \|\Psi a \|_\infty \leq \|\Psi b \|_\infty$ for elementwise nonnegative $\Psi$s?

Denote by $\Psi \in \mathcal{P}$ the property that $\Psi$ has non-negative entries and independent columns. Does the following property hold for $a,b \in \mathbb{R}^n$. $(\forall i. |a_i| \leq |b_i|)...
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144 views

Does isomorphism between normed space implies equivalence of norm?

Let $E$ be a vector space, $\|\cdot\|_1,\|\cdot\|_2$ be two norms on $E$, if $T:(E,\|\cdot\|_1)\to (E,\|\cdot\|_2)$ be an isomorphism (linear bijection, $T,T^{-1}$ are bouned), then does this imply ...
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How to prove that 2-norm of matrix A is <= infinite norm of matrix A

Now a bit of a disclaimer, its been two years since I last took a math class, so I have little to no memory of how to construct or go about formulating proofs. My tutor unfortunately is very and ...
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2k views

How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
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114 views

Showing coercivity of a function

I am well attuned to the definition for a function to be coerce, which is that $\lim_{\|x\| \to \infty}f(x) = \infty$ ie the values of $f$ go to infinity as the norm goes to infinity. So Ex.1 $f(...
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1answer
96 views

Question about a proof in Rudin's book - annihilators

I'm reading the proof of the following theorem in Rudin's "Functional Analysis": Let $M$ be a closed subspace of a Banach space $X$. The Hahn-Banach theorem extends each $m^* \in M^*$ to a ...
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49 views

value of the norm of the trace mapping

1) What is the exact value of the norm of the trace mapping ${\rm tr} \colon M_n \to \mathbb{C}$ where we equip $M_n$ with the operator norm $\|A\| = \sup\{\|Ax\| : x\in \ell^2_n \mbox{ with }\|x\|= 1\...
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53 views

Prove $\|\cdot\|_s$ is a norm, and find $m,M>0$ such that $m\|x\|_\infty\leq \|x\|_s\leq M\|x\|_\infty$

Here is my question - Let $\|\cdot\|_s:\mathbb{R}^2\to\mathbb{R}^2$ be defined by: $$\|(x_1,x_2)\|_s=\left\{ \begin{array}{l l} \|(x_1,x_2)\|_2 & \quad \text{$x_1x_2\geq 0$}\\ \|(x_1,...
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36 views

$p$-norm on $\mathbb{R}^n$ question

How I can show that $$\lim_{p \to \infty} \|x\|_{p} = \max\{|x_1|, \; |x_2|, \; \cdot ,\; |x_n|\}$$ if $\mathbb{R}^n$ has the p-norm? $p > 1$ of course. Has anyone done this or know how to? I'm ...
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306 views

Norm of the Linear (Integral) Operator on $C[0,1]$

There may be some similar questions being asked but I didn't see anything that was quite what I was looking for so I'll ask the question here. Let $X = (C[0,1],||.||_{\max})$ and $T:X \to X = \int^{t}...
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29 views

Help proof regarding sequence in subset of Hilbert space

I'm to prove the following: Let $H$ be a Hilbert space, and let $M$ be a non-empty convex subset of $H$. Suppose that $(x_n)$ is a sequence in $M$ such that $ ||x_n|| \to d$, where $d= \inf \{||x||:...
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152 views

Show that $\infty$-norm and $C^1$-norm are not equivalent.

Show that $\infty$-norm and $C^1$-norm are not equivalent. For the $C^1([a,b],\mathbb{R})$ space, show that $\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)|$ and $\displaystyle ||g||_{C^1}=...
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130 views

Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
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49 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 ...
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57 views

Matrix one-norm and infinity-norm

Help me please to find $3\times 3$ matrices $A$ and $B$ under following conditions: $\left \| A \right \|_{\infty }=4\left \| A \right \|_{1}, A \neq 0$ $\left \| B \right \|_{1}=4\left \| B \right \|...
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85 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
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42 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 inequality....
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53 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. $\psi(x)=\sum_{i}a_{i}T_{i}(...
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125 views

Immediate consequence of the definition of Operator Norm. Explain

||Av|| $\leq$ ||A||$_{op}$||v|| for every v $\in$ V I was wondering why this is true. Wikipedia says it's an immediate consequence of the definition but I just do not follow. I am using the ...
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27 views

How to prove this inequality for operator and function

How to prove this? $\sum_{k=1}^{\infty}|(Tf)_k|^2\leq ||T||^2||f||^2$ where $T$ is an operator and a function $f$. $(Tf)_k$ is the $k$-th coordinate of Tf. Should this involve Cauchy-Schwarz or the ...
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156 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
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559 views

What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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44 views

An equation related to covariance matrix, square root of the matrix, and Euclidean norm.

How can I prove this equation: $${ ({ x }^{ T }\Sigma x) }^{ 1/2 }={ \left\| { \Sigma }^{ 1/2 }x \right\| }_{ 2 }$$ In which $\Sigma $ is a covariance matrix. I tried some numerical examples in ...
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How to calculate the norm?

We have the vector : $$ w=(1,3,5,1,3,5,\ldots,1,3) \in \mathbb{R}^{3k-1}, $$ and we want to calculate its norm $\|w\|$. Now I would like to know how the norm $\|w\|$ can be calculated.
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110 views

An inequality on matrix norm

Does inequality $\|A\|_2\leq \| |A|_m \|_2 $ hold for all square matrices $A$ ? Where $|A|_m$ is also a square matrix, defined as $|A|_m := [|a_i,j|]$. Two examples are provided for the case that ...
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171 views

minimum trace norm on the set of matrices with fixed diagonal entries

What is the min nuclear norm (sum of singular values) on all $n \times n$ matrices$A$ whose diagonal is fixed. i.e. $diag(A) = v$ Is it true that the diagonal matrix is a minimizer?
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33 views

The space $\mathcal{D}((0,T);V)$ and its norm/embeddings?

Let $V$ be a Hilbert space. Define $\mathcal{D}((0,T);V)$ to be the set of functions $u:(0,T) \to V$ such $u$ is compactly supported on $(0,T)$ and is a $C^\infty$ test function. What is the norm on ...
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52 views

Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow \...
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324 views

Spectral radius, second induced norm

In my textbook there are few facts left without any sign of a proof, which really bugs me, and I was thinking maybe someone can help me: $A\in \mathbb{R}^{m,n} \Rightarrow \ \|A\|_2 = \sqrt{\...
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56 views

Norm of operator $T_x(f) = f(x)$

Let $X$ be a normed vectorspace and $X'$ be the dual space of $X$. For $x \in X$ we can define $T_x: X' \to \mathbb F$ by $T_x(f) := f(x)$. This is indeed an operator in $X''$. I read that $\| T_x \| =...
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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 ...
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129 views

Frobenius Norm with Unitary Operators

For something I'm working on, I have a matrix $A$ with another matrix $U$ which is unitary ($U^*U = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\|$. Now, I can do this ...
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81 views

A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete

Suppose $E$ is a normed space of all continuous functions on $[0,1]$ with norm $\int_{0}^{1}|f(t)|dt$. Prove that $E$ is not complete I know that we must do is to find a Cauchy sequence of continous ...
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90 views

Can $||f|| = ||a||_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$

We all know that: Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $||f|| = ...
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311 views

Condition Number of a Product

Is this hypothesis true? $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where $\operatorname {cond}$ is the condition number. And is this true for rectangular matrices? ...
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4k views

Proof of matrix norm property: submultiplicativity

I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define ...
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91 views

Prove norm inequality: $\|\mathrm x\|_2 \le \|\mathrm x\|_1$

On $\Bbb R^n$, define for $\mathrm x = (x_1, x_2, \ldots , x_n)$ a norm $$\|\mathrm x\|_1 := |x_1| + |x_2| + \cdots + |x_n|$$ By denoting the usual norm by $\|\mathrm x\|_2$, show that $\|\mathrm x\|...
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856 views

Convexity of the squared Frobenius norm of a matrix

I was reading this paper where the define an optimization problem as where K and L are kernel matrices and $\pi$ is the permutation matrix. They have explained that the function is convex because ...
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68 views

Complex analysis 2: $f \in \mathcal{H}(U,F)$

I have a problem: Suppose $U$ is an open set in $E$ and $f \in \mathcal{H}(U,F)$. Prove that: $1/.$ If $U=E$ then $r_bf(x)=\infty, \forall x \in U$; $2/.$ If $U \ne E$ then $r_bf(x)< \infty, ...
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82 views

$H^1$ function with smallest seminorm

Let $\Omega\subset\mathbb R^d$ be a Lipschitz domain and $u\in H^1(\Omega)$. Find a $w\in H^1(\Omega)$ with the same boundary values but minimal seminorm on $H^1(\Omega)$. I've read that harmonic ...
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775 views

Calculating the Norm of an operator in $L^2(0,1)$

If I have the following operator for $H=L^2(0,1)$: $$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that ...
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13 views

Positivity of a map in $(l^\infty(X))^*$

Let $X$ be a set and $\varphi: l^\infty(X)\to\mathbb{R}$ be a linear map such that $||\varphi||=1$ $\varphi(1_X)=1$ I am trying to prove that $\varphi(f)\ge 0$ for all $f\ge 0$, but all my ...