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

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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 ...
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66 views

Vector norms on $\Bbb R^n$

Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ be two vectors in $\Bbb R^n$. Suppose $\left|v_i\right|>|u_i|$ for all $i$. Let $\| \cdot\|$ be any vector norm on $\Bbb R^n$. Is it true ...
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913 views

show operator norm submultiplicative

We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and ...
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157 views

Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?

Let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $\ell^1$, suppose $x=\{x_n\}\in\ell^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$. Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
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48 views

Does $\|z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$ define a norm on $\mathbb R^2$

Does $\| z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$, with $z=(x,y)\in\mathbb R^2$, define a norm on $\mathbb R^2$?
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43 views

Question about norms

Denote supnorm by $|x| = \max\{|x_1|,...,|x_n|\}$ where $x \in \mathbb{R}^n$. How can we show that this norm and Euclidean norm satisfies the following inequality? $$ |x| \leq ||x|| \leq ...
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124 views

My “wrong” comparison between $\ell^2$ and $\ell^1$

For sure, $\ell^2$ is larger than $\ell^1$, because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$ But using Cauchy-Schwartz inequality, I get a "wrong" comparison: ...
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284 views

Is the problem of calculating the induced norm *difficult*?

Is the problem of calculating the induced norm of a linear operator (in a finite or infinite-dimensional space) generally a difficult one ? And by difficult I mean, that there are no closed formulas ...
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545 views

Poincaré inequality in unbounded domain

Help me please, how can I to show that Poincaré inequality in unbounded domain doesn't holds? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
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204 views

Exponential norm?

Can a norm "grow exponentially"? Let $||\cdot||_*: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0} $ be a norm such that: $$ \lim_{|x| \rightarrow \infty } \frac{ ||x||_* }{ e^{|x|} } > 0 $$ where ...
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Inequality involving norm of matrix integral

This question seems basic but I could not find an answer. I have seen the inequality $$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$ where $x(t) \in \mathbb{R}^n$ is a ...
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193 views

Maximum 1-norm of matrix times unit vector

If $A$ is a $p \times p$ matrix, what is $$\max_{||u||_2=1} ||Au||_1 ?$$ I am specifically interested in the case when $A$ is positive definite.
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25 views

Unit ball for a special norm

What does the unit sphere for the norm on $\mathbb{R}^2$, $\displaystyle N(x,y)\rightarrow\sup_{t\in\mathbb{R}}\frac{|x+ty|}{t^2+t+1}$, look like ? My approach was to consider $y=ax$ so as to get ...
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12 views

Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where ...
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34 views

Projection function preserving distance

Given $n>1$. I wis to construct a function $f:\mathbb{R}^n\mapsto\mathbb R$ such that $$\|X_1-X_2\|\le\|Y_1-Y_2\|\implies|f(X_1)-f(X_2)|\le |f(Y_1)-f(Y_2)|$$ for all $X_1,X_2,Y_1,Y_2\in\mathbb{R},$ ...
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48 views

The supremum norm is submultiplicative

Is the following proof correct: Let $X$ be compact a compact Hausdorff space and $C(X)$ the continuous functions $f: X \to \mathbb{C}$ on X. We can equip $C(X)$ with the (edit: sorry, semi-)norm ...
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40 views

Why is $L_0$ norm not convex? [closed]

I have this confusion in understanding the convexity of the $L_0$ norm. Why is $L_0$ norm not convex?
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71 views

Are the following norms equivalent?

We have the norms $||f||_1=||f||_\infty+||f'||_\infty$ and $||f||_2=|f(a)|+||f'||_\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I prove/disprove this.
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32 views

Norm of a Matrix-vector product

Suppose I have vector $\vec x \in \mathbb R^n$ and matrix $\mathbf M$ of dimension $m\times n$. Is there an alternative expression for $\lVert \mathbf M \cdot \vec x \lVert$ that includes $\lVert \vec ...
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42 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 ...
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54 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 ...
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127 views

L1-norm minimization

This is undoubtedly a trivial question but might as well ask: Why is the L1 norm minimization a heuristic for finding the sparsest vector? What I mean is that if the L1 norm sums the elements of a ...
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58 views

Can a vector space be complete for two non-compatible norms?

Let $V$ be a vector space and suppose that we have two non-compatible norms on it, i.e. I distinguish $E = (V, \|\cdot\|_1)$ from $F = (V, \|\cdot\|_2)$ and I ask that $\not\exists C>0 \; ...
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94 views

Matrix norm equivalence

If we define $ \|A\| = \max \{|A\cdot \mathbf{t}|:|\mathbf{t}|\leq 1\}.$ is it the same as defining it as $\max \{|A\cdot \mathbf{t}|:|\mathbf{t}|= 1\}$ ? If so, why? The book I'm following uses the ...
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75 views

Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
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45 views

Inequality of scalar-product and norm

Why does the following inequality hold, given $A$ is symmetric and $\lambda_{\min} (A)$ is the smallest Eigenvalue of $A$? $$v^\top A v \ge \lambda_{\min} (A) \; ||v||^2$$
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1k views

Frobenius Norm Triangle Inequality

How can I go about proving the triangle inequality holds for the Frobenius norm? I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
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472 views

Frobenius Norm Properties

Say $v\in {\mathbb{R}^{l}}$ (a column vector) and $A\in {\mathbb{R}^{l\times l}}$. I have encountered the following equality: $||v{{v}^{T}}-A||_{F}^{2}={{\left( {{v}^{T}}v ...
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256 views

Condition Number of a block Matrix

Is this hypothesis true? $$cond([A,B])≤cond(A)+cond(B)$$ where $cond$ is the Condition Number. And is this true for rectangular matrices($nxm$)? Let's consider $3$ different conditions for $A$ and ...
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84 views

Equivalency two norms?

Suppose that the following norms on $C^1[0,1]$ . Are they equivalent norms? $\|f\|=\|f\|_\infty+\|f'\|_\infty$ and $\|f\|=\max\{\|f\|_\infty, \|f'\|_\infty\}$ such that $f\in C^1[0,1]$ , ...
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421 views

Proving the triangle inequality for the L-2 norm $||x||_2 = \sqrt{x_1^2+x_2^2\ldots+x_n^2}$

I want to prove the triangle inequality for the l2-norm $||x||_2$: $$||x||_2 = \sqrt{x_1^2+x_2^2+\ldots+x_n^2}$$ $$\begin{align} \sqrt {\sum\limits_{i = 1}^n {{{\left( {{x_i} + {y_i}} \right)}^2}} ...
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557 views

Property of Subordinate Matrix Norm: $\|AB\| \leq \|A\|\|B\|$

I do not understand why the following property for Matrix subordinate norms holds: \begin{equation} \|AB\| \leq \|A\|\|B\| \end{equation} Please explain clearly as I know that it should be shown by ...
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490 views

Zero “norm” properties

I have seen the claim that the l0-norm ($\|X\|_0$ = support(X)) is a pseudo-norm because it does not satisfy all properties of a norm. I thought it to be triangle inequality, but am not able to show ...
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677 views

Proving that Euclidean space having the infinity metric is a complete metric space (stuck)

I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space. I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
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76 views

product of bounded linear operators

If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
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123 views

Norm of a function, Smoothness Penalization

I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
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74 views

Operator norm converging to 0 for certain condition

Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
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649 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 ...
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78 views

Two norms on $C_b([0,\infty])$

$C_b([0,\infty])$ is the space of all bounded, continuous functions. Let $||f||_a=(\int_{0}^{\infty}e^{-ax}|f(x)|^2)^{\frac{1}{2}}$ First I want to prove that it is a norm on $C_b([0,\infty])$. The ...
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85 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 ...
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What is the difference between the operator norm and the Euclidean norm?

I am quite confused whether the operator norm is the same as the Euclidean norm (2-norm). I know that :$\left \| A \right \|=\sup_{x\neq 0}\frac{\left \| Ax \right \|}{\left \| x \right \|}$. In ...
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Why one would want to normalize a matrix by dividing it by its Frobenius norm?

I am currently reading a scientific paper about clustering of brain signals, which consist on long time series across many channels (each signal is a matrix of C channels by T time samples). In the ...
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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\|= ...
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27 views

Some Norms on $V=P_n(\mathbb{R})$?

$V=P_n(\mathbb{R})$ be the vector space of all polynomials with degree $\le n$. I need to know Which of the followings are norms on $V.$ $\forall p\in V$ 1.$\|p\|^2=|p(1)|^2+\dots+|p(n+1)|^2$ ...
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31 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{$x_1x_2\geq 0$}\\ ...
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66 views

Convergence as for the norm [duplicate]

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $\|f_n\|_p \rightarrow \|f\|_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to ...
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24 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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50 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 = ...
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20 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 ...
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29 views

Inequality $\vert x^{*}y\vert\le \Vert x\Vert_1\Vert y\Vert_{\infty}.$

I would like to prove that the following exercise : For all $x,y\in \Bbb{K}^n$ we have $$\vert x^{*}y\vert\le \Vert x\Vert_1\Vert y\Vert_{\infty}.$$ Where $\Bbb{K}=\Bbb{R}$ or $\Bbb{C}$ and ...