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

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Existence of a vector whose norm is 1 in a Banach space

Given a finite dimensional Banach space V over reals, I have to show that there exists $v \in V$ such that $\|v\|=1$ At first, I thought that there's an identity element I in V. And $\|I\|=1$. But ...
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When is a norm of identity one?

Is there a specific condition that makes a norm (any norm) of identity equal to one in any Banach spaces? Thanks.
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Trace and Spectral norm of a matrix

Let $A_{n\times n}$ be a matrix. How I can show $$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$ and if $A$ is symmetric and positive definite, ...
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33 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
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33 views

Operator matrix norm associated with the vector norm

What is the operator matrix norm associated with this vector norm? $$\Vert x \Vert = \frac{1}{n} \mathop {\sum} \limits_{j=1}^{n} \vert x_j \vert, \qquad (x \in \mathbb{R^n})$$
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44 views

Is it possible that $\Vert I \Vert > 1$ !?

For any matrix norm, is it possible $\Vert I \Vert > 1$ ?, where $I_{n\times n}$ is identity matrix. If not, why in some books they write $\Vert I \Vert \geqslant 1 $ ?
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$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
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40 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...
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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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45 views

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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27 views

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
46 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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33 views

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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20 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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15 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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26 views

Differentiation of second norm

Considering $f = {e^{ - {{(||x - y|| - R)}^2}}}$ where ||.|| denotes the second norm, and $x \in {R^n}$. The following derivative is correct? $\frac{{\partial f}}{{\partial x}} = - 2(||x - y|| - ...
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1answer
18 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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26 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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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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18 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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31 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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21 views

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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15 views

The meaning of equivalence in norm.

Ask a elementary question: In WIKI: " However all these norms are equivalent in the sense that they all define the same topology." I think "these norms" here mean $l_1, l_2,...l_{inf}$ norms. ...
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19 views

On norms for “more complicated objects”

Whilst studying norms (via various routes) I was always confused by the following definition for some object $\phi$, $$\|\phi\|_p=\sup_{x\in A,x\neq 0}\frac{\|\phi x\|_p}{\|x\|_p}=\sup_{x\in ...
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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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35 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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39 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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29 views

Derivative of a Radial Basis Function $\nabla_x \frac{1}{1+\|x - y\|^d}$

I want to calculate the following derivative (with respect to x): $\nabla_x \frac{1}{1+\|x - y\|_2^d}$ where, $x,y \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector.
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equivalence of norms vs equivalence of absolute values

Consider a field, $k$. A norm on $k$ is a map, $|-| : k \to \mathbb{R}$ such that $(1)$ $|v|= \iff v=0$ $(2)$ $|vw| = |v||w|$ $(3)$ $|v+w| \leq |v|+|w|$ and we say two norms $|-|_1, |-|_2$ are ...
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How can I prove this operator is not continuos

Let $X$ the normed space of all polynomials on $J=[0,1]$ such that $||x||$=max$|x(t)|$ $t \in[0,1]$ and we have the following operator $Tx(t)=x'(t)$ prove this operator is not continuous
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Relation between max element norm and other norms of a matrix

I am looking for a relation between max element norm and other norms like $2$-norm or $\infty$-norm (in terms of bounds) so that I can reduce the following condition $$ ...
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42 views

How can I prove the following operator is continuous

Let we have the following operator $T$ From $C[a,b]$ to $C[a,b]$ by the formula $$Tx(t)=\int_{a}^{t}x(τ)dτ$$ How can I prove $T$ is continuous operator ?? The space $C[a,b]$ is squibbed with the ...
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20 views

Assuming $A$ is a nonexpansion in some norm, in what norm is $A^\top$ a nonexpansion.

Consider a matrix $A \in \mathbb{R}^{n \times n}$. Consider the vector norm $\| \cdot \|_\triangle = \| F \cdot \|_1$, where $F \in \mathbb{R}^{n \times m}$ and we have $m < n$ and $F$ has ...
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Is this inequality for finite sequences of numbers true?

Let $a_1,...,a_d$ be some positive reals with $\sum_{i=1}^d a_i \leq 1$. Let $x^k = (x^k_1,...,x^k_d)$ for $k=1,...,n$ be some vectors in $\mathbb{R}^d$. Is the following inequality true? And how is ...
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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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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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43 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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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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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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41 views

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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the spectral norm of the subtraction of two diagonal matrices

I saw this equation in some papers (e.g., http://cs-www.cs.yale.edu/homes/mmahoney/pubs/l2sample.pdf ), but I am wondering if it is wrong. The original equation is where $\|\|_2$ is the spectral ...
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26 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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28 views

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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12 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 ...