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

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Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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Upper bound for norm of matrix (cf. Example 2.7-7 in Erwine Kreyszig's book)

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the norm space of all ordered $n$-tuples of real numbers with the norm defined as ...
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is $N(f)=\int_{0}^{1} |f(t)|dt$ a norm on $E$(set of all continous real valued functions defined on [0,1])?

there are three conditions for a norm, I am stuck on one of them which is : $N(f)=0$ iff $f=0$. If i say $\int_{0}^{1} |f(t)|dt=0$ does this imply that $f=0$(zero function) ? Ok the other sense of ...
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Why does $(\vec{v}\cdot\vec{u})\vec{u}\neq(\vec{u}\cdot\vec{u})\vec{v}$?

I'm trying to save time running Gram-Schmidt. Why doesn't this product equal $||\vec{u}||\vec{v}$? More specifically (and I know this is fundamental and that I should already know it), why doesn't the ...
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1answer
140 views

Holder's inequality with expectation norm

One defines the ``p-expectation norm" of a function $f$ as $\vert f \vert_p = (\mathbb E ( f^p) )^{ \frac{1}{p}}$. What is the intuition for this norm? Now why are the following true? $\vert ...
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Frobenius Norm to L2 norm Problem

Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that $$ ||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2. $$ I am having ...
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What is the norm of the pre-multiplication by a fixed matrix operator?

Let $A \colon= \left(\alpha_{ij} \right)_{m\times n}$ be a given $m \times n$ matrix of complex numbers, and let the operator $T \colon \mathbb{C}^n \to \mathbb{C}^m$ be defined by $$T(x) \colon= Ax ...
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92 views

Compute the value of the Sobolev norm in $H^{-1/2}$

I am working with finite elements using domain decomposition in 2D and one of the solutions I need to obtain is the co-normal derivative of the solution along a segment that is the intersection of two ...
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29 views

Infinity Norm calculation $\| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|$

I have the following infinity norm: $$ \| \ln(x) - (\ln(3/2) + 2/3(x-3/2)) \|_\infty. $$ Computing from [1,2]. I know that I can compute this in matlab and I get .072. However, how would one go ...
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75 views

Norm of orthonormal basis

I know that an orthonormal basis of a vector space, say V is a orthogonal basis in which each entry has unit length. My question is, then, if you have some orthonormal basis say $\{v_1,\ldots,v_8\}$ ...
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1answer
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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108 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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134 views

How would I calculate the error in LU Decomposition of a matrix?

I am currently practicing the LU Decomposition of n x n matrices. After finding the LU decomposition, I am lost on how to find the error. I am trying to understand the notation in a problem that asks ...
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48 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
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Why is this an nuclear norm?

The following figure is from a paper: "The Convex Geometry of Linear Inverse Problems" Just on p.3. As in the figure, the red lines are $2 \times 2$ symmetric unit-Euclidean-norm rank-one ...
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1answer
35 views

Norm of a difference of two elements in Quadratic Fields

everyone. Hope you can help me out. I ran across this question and have absolutely no idea how to go about: "Let $X =x+y\sqrt{2}$ with $x,y\in\mathbb{Q}$. Show that there is an element ...
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Proving an inequality involving norms of real functions.

If $ f : [a,b] \subset \mathbb{R} \rightarrow \mathbb{R} $ is continuous and differentiable in $(a,b)$, then one can define a norm for such functions as $$ \|f\| = |f(a)| + \max_{x \in (a,b)} |f ...
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42 views

Functional analysis, normed space

Can you please help me with this exercise.. I have to check if in interval $[a,b]$ continuously differentiable functions $x=x(t)$ norm can be defined as: $$\vert x(b) - x(a) \vert + \max_{a \leq t ...
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Can one find a stronger norm on a Banach space?

Given a Banach space $V$ of infinite dimension with norm $\|\cdot\|_1$, is that possible to find a norm $\|\cdot\|_2$ on $V$ such that the topology induced by $\|\cdot\|_2$ is strictly stronger than ...
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161 views

Proof of relation between maximum element and induced $p$-norm of a matrix

If true, prove the identity: $$ ||A|| \ge \max\limits_{i,j}|a_{ij}| $$ $||.||$ is any induced/operator norm. Edit: The identity is true only for operator norm induced by $p$-norm for vectors. I ...
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How can I prove this operator is not continuos

Let $X$ be the normed space of all polynomials on $[0,1]$ such that $\| x \| = \max \limits _{t \in [0,1]} |x(t)|$ and we have the following operator $Tx(t)=x'(t)$. Prove this operator is not ...
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How can I prove the following operator is continuous

Let us have the following operator $T : C[a,b] \to C[a,b]$ by the formula $$Tx(t)=\int \limits _a ^t x(τ)\ \Bbb dτ$$ How can I prove $T$ is a continuous operator? The space $C[a,b]$ is equipped with ...
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About a norm : $p(uv)=p(u)p(v)$ all the time? [closed]

Say, $p$ is a norm on a vector space. Then can we say that $$p(uv)=p(u)p(v)$$ all the time? Thanks.
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Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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21 views

Mahalanobis distance to the ellipsoid center

I am confused about the following description: If we parameterize the ellipsoid $E$ as: $E = \{x|\ ||Ax-b||_2 \leq 1\}$. $A \in S_{++}^n$ Then the Mahalanobis distance to the ellipsoid center is ...
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90 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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26 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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70 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
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Question about an inequality in a published paper which is yielded from an approximation

I am reading a published paper on K- SVD: An algorithm for designing overcomplete dictionaries for sparse representation In the introduction, it says: Recent years have witnessed a growing ...
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29 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 ...
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Topology induced by norm

What is the meaning of topology induced by norm. To me topology is a collection of subsets satisfying certain rules. How can a norm induce a topology...? For example how can $\|\cdot\|_{2}$ induce a ...
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14 views

Bicomplex quadratic forms

The quadratic form associated with a real matrix $Q$ and a real vector $\vec{x}$ is $\langle Q \vec{x}, \vec{x} \rangle$. The quadratic form associated with a real matrix $Q$ and a complex vector ...
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Prove that the norm in an inner product space is $ \ge 0$

Macdonald Linear and Geometric Algebra defines an Inner Product Space in the following way (pg 57): "An inner product space is a vector space with a product called an inner product. The inner product ...
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Question about the norm $\|A \cdot \|_1$

$A \in \mathbb{R}^{n \times m}$, where $n > m$, has independent columns. Consider the vector norm $\| x \|_\Box = \| A x \|_1$. Consider two matrices $M_1$, $M_2$ which preserve this norm, i.e. ...
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Dimensionality of tangent vectors in R^2

I am puzzled with the following problem: given a tangent vector (a d/dx) in the Euclidean plane R^2 with "a" a dimensionless scalar, the dimensionality of this vector is, I suppose, 1/[lenght] and ...
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Why doesn't the $L_2$ norm differentiable at $x=0$?

Why doesn't the $L_2$ norm differentiable at $x=0$? Let's define $N(x)$ as the norm function. I know that for every $x\ne 0$: $$\frac{\partial N}{\partial x_i}(x) = \frac{x_i}{\|x\|}$$ What ...
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1answer
47 views

Determining Infinity Norm

I am trying to follow along with a practice problem in my book that asks me to determine the distance between two vectors $\textbf{u} = (-2, 2, 1)^{T}$ and $\textbf{v} = (1, 4, 1)^{T}$ using the ...
2
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1answer
81 views

Show that $F(x) = f(\|x\|)$ is differentiable on $\mathbb{R}^n$. [duplicate]

Let an even function $f:\mathbb{R}\to\mathbb{R}$ which is even and differentiable. We define $F:\mathbb{R}^n\to\mathbb{R}$ as $F(x) = f(\|x\|)$. Show that $F(x)$ is differentiable on ...
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62 views

Rank as norm on matrix

Could we consider matrix rank $r$ a norm? Is other norm similar to rank $r$ possible to associate with a finite matrix? (We denote trivial valuation $|\alpha|=1,\quad\forall\alpha\in\Bbb F^*$ where ...
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Bound on Symmetric Matrices

Let $A=(a_{ij})$ be a matrix with real entries, $1 \leq i, j, \leq n$. Let $A^{T}=(a_{ij}^{T})$ be the transposed matrix, that is $a_{ij}^{T}=a_{ji}$. Suppose that $a_{ij}=a_{ji}$, namely $A$ is a ...
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Hessian of Frobenius norm

I want to find the Hessian of the following function, $F(\mathbf{X}) = \frac{1}{2}||\mathbf{Y} - \mathbf{AX}||_F^2$. My try: Using trace formula for Frobenius norm, $F(\mathbf{X})$ can be written as, ...
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Why is $\ell_{2,1}$ mixed norm non-smooth?

I'm reading about optimization problems involving mixed norms. In particular I'm getting acquainted with the $\ell_{2,1}$ norm. For a matrix $\mathbf{X}$, the $\ell_{\alpha,\beta}$ norm, ...
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Equivalence of norms between two Banach spaces

I just learnt open mapping theorem. And I met a statement online asserting that If $X$,$Y$ are Banach space, and $T:X\to Y$ be a continuous bijection, then norms for $X,Y$ are equivalent. Can we ...
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Norm of the product of an isometry and a bounded operator

Let $A$ be a bounded operator and $V$ a linear isometry, both defined on a complex Hilbert space $H$ (infinite dimensional). I could easily prove that $\|VA\|=\|A\|$. But, I just couldn't prove that ...
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Squaring Norms solved by Algebra

I found the following in a paper and am not sure how it is correct: $\Vert a - b \Vert^2$ was expanded to: $\Vert a \Vert^2 - 2a^Tb + \Vert b \Vert^2$ The paper was on gps location algorithms so ...
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32 views

Gradient of this function

I am to solve an optimization problem as described below: $$ \min f(x) = \frac{1}{2}\left\lVert x - x_{b} \right\rVert^{2}+ \frac{1}{2}\left\lVert \epsilon \right\rVert^{2}$$ with $$ Hx -y = \epsilon ...
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why and when t0 use norm instead of abs and vice versa

What is the difference between the norm and abs of an expression.. as far i understand does ||a - z|| mean norm and |a-z| abs , but what is the difference?
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Norm controls the components of a matrix- proof?

For certain norms (such as the Frobenius norm) it is clear that the norm provides component-wise control - each component is at most as large as the norm in magnitude. How do we establish this for ...
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39 views

Apply Cauchy-Schwarz to vector?

For $x_i, y_i \in \mathbb{C}$, the C-S inequality gives $$\left| \sum_{i=1}^n x_i \bar{y}_i \right|^2 \leq \sum_{j=1}^n |x_j|^2 \sum_{k=1}^n |y_k|^2 .$$ Is it true if $x_i, y_i$ are actually ...