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

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Lipschitz condition normed vector space

Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition? Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
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513 views

What is the role of supremum in operator norm

An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
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81 views

Linear functional $\mathscr{L}(E,F)$

Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$. Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question: How to prove ...
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94 views

Study the equivalence of these norms

I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique. So I define the ...
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197 views

On norm selection for the solution of an overdetermined linear system

I am considering the following linear system: $Ax = b$ Where: $A$ is $9000 \times 139$ $x$ is $139 \times 1$ and sparse $b$ is $9000 \times 1$ Most of the resources I have found online point to ...
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45 views

Order of infinite dimension norms

I know that $$\|{f}\|_{L^1(0,L)}\leq\|{f}\|_{L^2(0,L)}\leq\|{f}\|_{\mathscr{C}^1(0,L)}\leq\|{f}\|_{\mathscr{C}^2(0,L)}\leq\|{f}\|_{\mathscr{C}^{\infty}(0,L)}$$ But I don't know where to put in this ...
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87 views

Determining a norm from a quadratic form

If $B$ is a quadratic form over some space $V$, what is the norm determined by $B$? Is this the inner product $\langle Bu,Bv\rangle$? If not, and it is not possible to determine a norm from knowing ...
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117 views

Does π depends on the norm? [duplicate]

If we take the definition of π in the form: π is the ratio of a circle's circumference to its diameter. There implicitly assumed that the norm is Euclidian: \begin{equation} ...
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71 views

A bound on the norm of the sum of two index-disjoint matrices

Given two matrices, it is well known that $\parallel A+B \parallel _2 \leq \parallel A \parallel _2+\parallel B \parallel_2$. Now, suppose that the nonzero indices are disjoint (i.e., $A$ is nonzero ...
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495 views

Change in singular values of matrix after left-multiply with a diagonal matrix

Say that we have an SVD for a matrix $X = U \Sigma V^T$, giving trace norm $||X||_{tr} = ||\Sigma||_{tr} = \sum \Sigma_{ii}$. I am wondering what happens to the SVD and/or trace norm if we left ...
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186 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
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90 views

Norm of Polynom

I'm trying to understand the following equation: $$\langle p,q\rangle_p=\int_{-1}^1p(x)q(x)dx\\ \text{Basis: }\{p_1,p_2\}\\ p_1:=2x,\quad p_2:=x-1$$ Norm: $$p_1:q_1=\frac{p_1}{\|p_1\|_p}$$ ...
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131 views

Matrix norm less than $1$ iteration

Is the following true always for a matrix norm $$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$ Related to this given $r$ is positive constant, $H$ is symmetric positive ...
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141 views

Small question regarding norms and Holder conjugates.

I'm trying show that if $p,q$ are Holder Conjugates then: $$\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left<a,x\right>$$ Where ...
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prove matrix norm equivalence

Given $A \in R^{m\times n}$, I need to prove: $$||A||_2 \le \sqrt {m}||A||_\infty$$ I have tried a number of things and I just cant seem to get it to work. Also, I need to prove: $$||A||_2 \le ...
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73 views

Does switching between different $L_p$ norms preserve order?

Suppose you have a list of $n$ dimensional vectors. One can order them by using an $L_p$ norm to do comparisons between vectors. The general questions is, will the order be different depending on the ...
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103 views

Estimating the integral in norm.

I want to estimate the integral $$\int k(x,y)f(y)dy$$assuming the fact that $k(x,y), f(y)$ are in $L^p, L^q$ respectively. But I want to bound the the whole integral in $L^r$, $r\in [1,\infty]$. I ...
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114 views

How to calculate norm of operator in Hilbert space

LEt $H_1,H_2$ are two Hilbert spaces. $\{e_1,\ldots,e_n\}\subseteq H_1$ and $\{f_1,\ldots,f_n\}\subseteq H_2$ two orthonormal systems. $\lambda_1,\ldots,\lambda_n\in\mathbb K$. Let $$ U:H_1\to ...
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73 views

Could Ky-Fan Norms Improve the Bound on the Max Norm of $A$

Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. I've learned that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$, where $\|A\|_{\text{max}}$ means the Max norm. I want to get a ...
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How are $\|A\|_{\text{max}}$ and $\|D\|_{\text{max}}$ related?

Max norm The max norm is the elementwise norm with $p = \infty$: $$ \|A\|_{\text{max}} = \max \{|a_{ij}|\}. $$ This norm is not sub-multiplicative. Let $A$ be real symmetric and $D$ ...
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195 views

essential supremum of a matrix multiplication operator

Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n) $ is a ...
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299 views

Equivalence of Schatten and spectral norms

I'd like some help showing the equivalence of these two norms when $p = \log n$. Recall the $p$-th Schatten norm of a linear operator $A$ acting on $\mathbb{R}^{n}$. In the particular case of $p = ...
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equivalence of two definitions of norm equivalence: “$|\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 ...
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195 views

Representing with Hilbert Schmidt Norm

Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals ...
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430 views

Time complexity of norm function in Matlab

What is the time complexity of the "norm" function in matlab?
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201 views

p-norm of series / Mathematica

When computing the $p$-norm of the series $x_n = \mathrm e^{\mathrm i n}$ for $n \in \mathbb{Z}$ (not $\mathbb{N}$), the sum should not converge: $$\|x\|_p^p = \sum_{n \in \mathbb{Z}} (\mathrm ...
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98 views

What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
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71 views

Equivalence of a Vector Norm being Absolute

I'm trying to show that a vector norm $\|\cdot\|$ being absolute ($\|x\| = \|\;|x|\;\|)$ is equivalent to showing that $\|x'\| = \|[\alpha_1x_1\ldots\alpha_nx_n]^T\| = \|x\|$ for all ...
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280 views

Help with a proof that the inversion mapping for matrices is continuous

I'm reading a proof that the map $A\mapsto A^{-1}$ is continuous in the operator norm. A part of the proof is that if $A,B$ are matrices such that $A$ is invertible and $\beta=\parallel ...
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28 views

$\operatorname{Im} A = (\operatorname{ker} A^*)^\perp$

Let $A:\mathbb{R}^m \to \mathbb{R}^n$ be a linear transformation. We know that there is a unique transformation $A^*:\mathbb{R}^n \to \mathbb{R}^m$ such that $$\langle Ax,y\rangle = \langle x,A^*y ...
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Euclidean norm bound

Consider the variable $x = (x_1,\ldots,x_n)$, where each element $i$ is restricted to live in the interval $\underline{x}_i\le x_i \le \bar{x}_i$. Now consider the norm $\|x\|_2$. Why can't I write ...
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An equivalent definition of the condition number of a matrix [on hold]

How can I prove that the condition number can't be expressed by $$\kappa(A)= \sup_{\lvert\lvert x \rvert \rvert=\lvert \lvert y \rvert \rvert} \lvert\lvert Ax\rvert \rvert/\lvert\lvert Ay\rvert ...
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Is there a name for this function with properties…

Let $V$ be a vector space over an algebraic structure $\mathbb{A}$, and suppose we have a binary operation $\star:V^2\to V$. Consider a function $f:V\to \mathbb{A}$ with the property that $$f(x\star ...
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Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
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comparison of norms in vector ODEs

Assume I have the following equation $$ \dot{f}(t) \le A(t)f(t),\quad f(0) = f_0$$ where $f:[0,\infty]\rightarrow\mathbb{R}^n$ has positive components ($\ge0$), ...
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How can we check that for a given norm, we can found an inner product?

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers. Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner ...
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63 views

Can I solve a problem like a combination of PCA and compressed sensing?

$$ \underset{A,x}{\text{minimize}} \quad \lambda \left\| x \right\|_{1} + \left\| A \right\|_{*} $$ $$ D = A + Mx $$ Where $M \in \mathbb{R}^{n \times m}$, $x \in \mathbb{R}^{m \times z}$, $E=Mx \in ...
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53 views

Completeness in $L^p$ and supremum norms

As seen by this question, $C([0,1])$ is not complete in the $L^p$ norm ($1 \le p < \infty$) because one can choose a Cauchy sequence $$f_n(x) := \begin{cases} 0 & \text{if $0 \le x \le ...
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28 views

Do tensor norms exist?

Does there exist norms for tensors, as an extension for the ordinary matrix norm? For example, if there is a derivative of a matrix [A] with respect to a vector {x}, does the norm of this derivative ...
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Compute the limit of this expression of norms:

Compute the limit, as n goes to infinity, of the quotient: $$\frac{||A^{n+2}(x)||}{||A^n(x)||} $$, given the matrix $$ \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ ...
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36 views

Absolute value of vector not equal to magnitude of vector

I've come accross the following inequality for a norm (where the norm defines the length of the vector): $$\lvert x \rvert ≤ \lvert \lvert x \rvert \rvert \leq \sqrt{n} \lvert x \rvert$$ where $x$ ...
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Determine interior and boundary of a set

Let $(X,||\cdot||)$ be a normed vector space, where $$X = \big\{ (a_n)_{n \geq 1} ~~|~~ (a_n)_{n \geq 1} \text{ is a bounded real sequence }\big\}$$ and $$\|(a_n)_n\| = \sup_{n \in ...
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convergence in different normed spaces

In my class lecture notes, there is such a lemma. Let $X$ be a vector space over $\mathbb F$ and $\lVert \cdot \rVert_1$, $\lVert \cdot \rVert_2$ be two norms on it. If there is $M > 0$ such that ...
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32 views

Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
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48 views

What kind of matrix norm satisifies $\text {norm} (A*B)\leq \text {norm} (A)*\text {norm} (B)$ in which A is square?

$||A\times B||\le ||A||\cdot ||B||$ is not always correct. But which kind of matrix norm satisifies this formula for square matrix $A$ and arbitrary matrix $B$?
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Holder Inequality when $0 < p < 1$ [duplicate]

If $0 < p < 1$, $f \in L^p$, and $\int \lvert g \rvert^q < \infty$, show that $$\int \lvert fg \rvert \ge (\int \lvert f \rvert^p)^{\frac{1}{p}}(\int \lvert g \rvert^q)^{\frac{1}{q}}$$ My ...
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76 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
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“Transference” argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
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condition to be a norm on a finite dimensional complex vector space

Here x is an element of C^n, where n is a natural number and C is the complex number field, and p is any positive number. This becomes a norm when p>=1. But, what happens when p is between 0 and 1? ...
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Order of two vectors to maximise the norm

Given vectors ${\bf a} = [a_1, \dots , a_n]^T$ and ${\bf b} = [b_1, \dots , b_n]^T$, a permutation $\pi$ acting on $[1, \dots ,n]$ and defining ${\bf b}^{\pi} = [b_{\pi(1)}, \dots , b_{\pi(n)}]^T$, ...