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

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48 views

Limit to zero of the $p$-norm

I have the $p$-norm defined as $$\|x\|_p=\left(\sum_{k=1}^n|x_k|^p\right)^\frac{1}{p}.$$ I am trying to find the limit as $p\to0^+$ of $\|x\|_p$. I've seen it defined as $\{x_k:x_k\neq0\}$. Why is ...
2
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1answer
38 views

$A\in \mathbb{R}^{n\times m}$, proove that the sum of the maximum of each row is less than $n\|A\|_1$

I'm quite sure this is true, but I'm having trouble in finding a proof for $\mathbb{R}^{n\times m}$. I already found it for $\mathbb{R}^{2\times 2}$. Let $A\in \mathbb{R}^{n\times m}$, such that: ...
2
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3answers
34 views

Hölder inequality in case $q=p=2$.

It should return the Cauchy-Schwarz inequality, but I'm having trouble with comparing the left sides of the inequalities: For example, if $x=(4,3)$ and $y=(3,-4)$, then $\sum_{v=1}^2 |x_vy_v| = ...
2
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1answer
58 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
2
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1answer
51 views

Inequality$\Big|\sum_{j=1}^n a_{1j} x_j \Big|^2 \leq \sum_{j=1}^n |a_{1j}|^2 \sum_{j=1}^n |x_j|^2$

Let ${\bf A}$ be a $m \times n$ matrix with entries $a_{ij}$, and ${\bf x}$ be a $n \times 1$ vector with entries $x_{i}$. Then how can I show $$ \left\vert\,\sum_{j\ =\ 1}^{n} ...
2
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1answer
40 views

Understanding an equality with open balls

In order to understand a proof I want to know why the following is true: Let X be a banach space and $x\in X$ with $x\in B(x_0,r)$ (open ball around $x_0$ with radius $r$), then $\frac{r}{2}x-x_0 \in ...
2
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1answer
81 views

Counterexample for $\| (\Phi^\top \Xi \Phi)^\dagger \Phi^\top \Xi P \Phi \|_2 \leq 1$

All matrices are real. Let $P$ be any stochastic matrix (i.e. square, non-negative, rows summing to one). Let $\Xi$ be a diagonal matrix containing on its diagonal any left eigenvector of $P$ ...
2
votes
1answer
79 views

Calculating a Matrix Norm

I'm trying to calculate some norm for a matrix $A = [3, 2; 0,1]$ given the formula $\|A\| = \max_{|v|=1}|Av|$, where $|v|$ is taken to be the Euclidean norm for a vector, i.e. the standard distance ...
2
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1answer
49 views

Norm on $\mathbb R^n$ with given unit ball

Consider a finite subset $S$ of $\mathbb Z^n$ such that $-s\in S$ whenever $s\in S$ and $S$ generates $\mathbb Z^n$. What is a norm on $\mathbb R^n$ whose unit ball is precisely the convex hull of ...
2
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1answer
38 views

$\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$

I want to prove this $\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$\ Suppose $\lambda_0>\lambda_1>\dots>\lambda_{n-1}\ge 0$ are distinct eigen values of $T^*T$ and ...
2
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1answer
77 views

Proof that the condition number of an isometry matrix is 1

At first glance it seems somewhat trivial, but I have some doubts, so I'd like your opinion. We are given that $\left\|Ax\right\| = \left\|x\right\|, ~ \forall x \in \mathbb{C}^{n}$ and want to show ...
2
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1answer
51 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
2
votes
1answer
63 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 ...
2
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1answer
54 views

Where does the definition of the $L_0$ norm come from?

Where does the definition of the $L_0$ norm come from? $$\|x\|_0=|S|$$ Where $S=\{x_k:x_k\neq 0\}$
2
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1answer
34 views

Geometric characterization of an Euclidean norm

Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse. I'm stuck on this one. I do not see how can I connect the definition of an ...
2
votes
1answer
301 views

Is the matrix least squares minimizer (Frobenius norm) the same as the matrix 2-norm minimizer?

Given matrices $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{n \times k}$, consider the (least squares) minimizer $\arg \min_{X \in \mathbb{R}^{m \times k}} \| AX - B \|_F$, where $\| M \|_F$ ...
2
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1answer
64 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 ...
2
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1answer
105 views

How to FAST calculate 2 norm / spectral norm of a matrix.

I meant reduced 2 norm, the largest singular value. My current approach is applying the SVD decomposition of A via "?gesdd" in MKL, and then taking the largest singular value. I think there should ...
2
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1answer
33 views

Find conditioning of the matrix

Find conditioning of the following matrix: $$A=\begin{bmatrix}1& 0\\1&\epsilon\end{bmatrix}.$$ in a $\|.\|_\infty$ norm for $\epsilon > 0$
2
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1answer
77 views

For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. ...
2
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1answer
144 views

bound on trace-norm of product of matrices

Is it true that $$ \|ABA^\dagger\|_1\leq \|A\|^2\|B\|_1, $$ where $\|A\|_1$ is the trace norm, $\|A\|$ is the spectral norm, and $A$ and $B$ are square matrices?
2
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1answer
59 views

Showing that a map defined using the dual is a bounded linear operator from X' into X'

I have trouble answering the second part of the following exercise. Any help would be appreciated! Let $(X, \| \cdot \|)$ be a reflexive Banach space. Let $\{ T_n \}_{n = 1}^\infty$ be a sequence of ...
2
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1answer
49 views

Representation in Banach space and norms 'induced' by representation

By $G$ we denote some compact group, $X$ stands for some Banach space. Suppose $\pi\colon G\longrightarrow \mathrm{GL}(X)$ to be some representation in $X$. I'm trying to prove that there is an ...
2
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1answer
34 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
2
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1answer
62 views

Derivative of function that includes norm

I was solving the problem: find the derivative of a function f : H → R, $f (x) = \sin ||x||^3$ (H is Hilbert space). I got the answer $f'(x)=3\cos||x||^3 x||x||$. Is this correct or I am doing ...
2
votes
2answers
452 views

Any two norms on finite dimensional space are equivalent

Any two norms on a finite dimensional linear space are equivalent. Suppose not, and that $||\cdot||$ is a norm such that for any other norm $||\cdot||'$ and any constant $C$, $C||x||'<||x||$ for ...
2
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1answer
79 views

Show that every operator norm is consistent

Is the following a correct way to show that operator norms are consistent? $$ \|AB\|=\max_{Bx \ne 0}\frac{\|ABx\|_\alpha }{\|x\|_\alpha} =\max_{ Bx\ne 0}\frac{\|ABx\|_\alpha}{\|Bx\|_\alpha} ...
2
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1answer
100 views

Abstract Algebra: Field extensions

I'm trying to prove the following but it causes me a lot of trouble: Let $L$ be a finite extension of degree $n$ of a field $K$ with characteristic $0$. Let $\sigma_1,\dots,\sigma_n$ be different $K$ ...
2
votes
2answers
203 views

Proof that two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent

Two norms $\def\norm#1{\lVert#1\rVert}\norm\cdot_1$ and $\norm\cdot_2$ are equivalent iff $\;\exists\;c_1,c_2>0$ such that $c_1\norm x_1\le \norm x_2\le c_2\norm x_1$ Show that $\norm ...
2
votes
1answer
75 views

Minimization of norms

How do I minimize the following? $ min_{z>0} - zt + 1/2\ z\ ||\ Y + X_k\ /\ z\ ||_2^2 $ Also, $X_k^TX_k = 1 \ \ \forall k $ I am given that the answer should be : $ \sqrt{Y^T - 2t} + Y^TX$ ...
2
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1answer
37 views

What could be the lead to prove $||X||_2 \leq ||X||_F \leq \sqrt{rank(X)}||X||_2$?

In the above statement, $||X||_2$ = $L_2$ norm of X and $||X||_F$ = $Frobenius$ norm of X. It appears to me that the $L2$ norm of X and $Frobenius$ norm of X are the same. How should i proceed to ...
2
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1answer
50 views

Help showing $\phi _k$ is a bounded linear functional

Let $V$ be the space of continuous functions on the interval $[-\pi , \pi]$ with the $L^2$ norm $$\lVert f\rVert_2=\left(\int_{-\pi}^\pi |f(t)|^2\mathrm dt)\right)^\frac{1}{2}$$ For $f$ in $V$, define ...
2
votes
1answer
104 views

How to force unitary Euclidean norm in a complex matrix by multiplication with a diagonal matrix

I need to solve the following problem: Suppose a non-sparse, non-singular complex matrix $\mathbf{P}$. If I want to force all rows in $\mathbf{P}$ to present unitary Euclidean norms by multiplying ...
2
votes
1answer
90 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
2
votes
1answer
522 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, ...
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1answer
1k views

Constrained infinity norm minimization

I have a problem like this: $$\min_x |Ax|_\infty \text{ s.t. } \sum_i x_i = c$$ That is, I want to find the vector $x$ whose elements sum to a constant $c$ that minimized the infinity norm of $Ax$. ...
2
votes
1answer
96 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...
2
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1answer
51 views

Regularity and the Varitational Inequality

Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
2
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1answer
115 views

Operator norm estimate

Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$. Furthermore, let $B\colon H\rightarrow C[a,b]$ be a bounded operator. According to the Riesz-Frechet theorem there is ...
2
votes
2answers
132 views

How to show for a PSD matrix $A$ that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$?

If $A \in \mathbb{C}^{n \times n}$ is positive semidefinite, show that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where $\sigma _{\min}\left ( A ...
2
votes
1answer
150 views

finding operator norm $T_N$

How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by $T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer . TIA
2
votes
1answer
322 views

Invertible operator norm bound

Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...
2
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1answer
994 views

Norm equivalence Sobolev space

I have this problem: Let $k>0$ (integer) and $1 \leq p < \infty$. Show that the norms $$ ||u||_{W^{k,p}(U)} = \bigg( \sum_{|\alpha|\leq k}||D^{\alpha}u||_{L^{p}(U)}^{p}\bigg)^{\frac{1}{p}} $$ ...
2
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1answer
187 views

Is the polar set of convex Polytope also Polytope

Let $P$ be a convex polytope. How can I prove that the polar set of $P$ (lets call it $P^*$) is polytope? where $P^*=\{x\in\mathcal R^n:\forall v\in P, |\langle x,v\rangle|\le1\}$ . $Thanks$
2
votes
1answer
158 views

What are the maximumnorm and supremumnorm of a vector when having a basis?

I have a perhaps stupid question. When having a finite-dimensional Vectorspace $X$ (f.e. n-dimensional) and when knowing a basis $V=\left\{v_1,...,v_n\right\}$ of it, so any $x\in X$ can be written as ...
2
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1answer
59 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
2
votes
2answers
2k views

convergence of $L^p $ norm [duplicate]

Possible Duplicate: Limit of $L^p$ norm If I define $|f|_{L^\infty}= \lim_{n\to \infty} |f|_{L^n}$. How can I prove that this limit is esssup $|f|$?
2
votes
1answer
277 views

Quotient norm on $X\backslash M$

I have $X=(C([0,1]),||.||_1)$ where $||f||_1=\int_{0}^{1}|f(t)|dt$ and $M=\{f\in C([a,b]): f(0)=0\}$. Now I have three questions: 1) Is the quotient norm a norm on the quotient space X\M ? What I ...
2
votes
1answer
675 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 $\lim_{n\to\infty}\|x_n-x\|_1=0.$ ...
2
votes
1answer
591 views

Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...