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

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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 ...
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how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
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42 views

Is convergence in the norm equivalent to convergence of norms?

If $\| \cdot \|$ is a norm on some space. Does the equivalence go both ways? $$\| f_n-f \| \to 0 \Longleftrightarrow \| f_n\| \to \| f\| $$ The $\implies$ direction is obvious since $\| f_n-f \| ...
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1answer
51 views

Integral Operator Theory on $L^2[0,1]$

Let K be the integral operator on l^2[0,1] defined by itex(t) = \int_0^t (t-s)f(s)\,ds[/itex] where 0\leq t\leq 1 Show that ||K|| <1 and that tex(t)= \int_0^t ...
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52 views

about vectors norm

in the following article http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf page 3 he say: $$y= \langle y , a_{k_0} \rangle a_{k_0} + R $$ with $a_{k_0}\in D$ with $\forall ...
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1answer
69 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
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26 views

Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...
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51 views

Check that: it is a norm

My question is: Let $f \in C^{1} [0,1]$ & let $f'$ denote its derivative. Define: $|| f ||_{1} = ( \int_{0}^{1} (|f(t)|^{2} + |f'(t)|^{2})dt)^{\frac{1}{2}}.$We are to show that: $||f||_{1}$ ...
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32 views

Show that the set of points that are nearer $a$ than $b$ with respect to $\lVert \cdot \rVert_2$ is convex

I am trying to show the above statement: Show that the set of points that are nearer $a$ than $b$ in the sense of Euclidean $\lVert\cdot\rVert_2$ are convex. My attempt This follows from the ...
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0answers
29 views

How to detect reflexivity of the closure

Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue $L_p$ norm is reflexive when $1 < p < \infty$, but it is not reflexive for $p = 1$. How ...
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34 views

A question on minimizing $\| . \|_2^2$ vs $\| . \|_2$

Suppose we are in $\mathbb{R}^n$ Is the problem of $d(x,Y) = \inf\{ \| x - y\|^2 : y \in Y\}$ equivalent to $d(x,Y) = \inf\{ \| x - y\| : y \in Y\}$ Pardon me, let us keep it simple and just stick ...
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1answer
41 views

Upper-bouding $||A||_2$ with $||A_i||_2$, where $A_i$ are rows of $A$

Let $A$ be an $n \times n$ matrix and $A_i$ be its $i$-th row. We know $||A_i||_2 \le c_i$ for each $i$, that is, the upper bound for each row. Now we want an upper bound for $||A||_2$ using the ...
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70 views

Parallelogram law in normed vectorspace withour an inner product.

Let $V$ be any $\mathbb{K}$-vectorspace with norm $\|\cdot\|$ I know that the Parallelogram law holds if the norm is induced by some inner product $\langle\cdot,\cdot\rangle$, i.e. $$ ...
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1answer
23 views

Check this is a hilbert norm: $ \ell^2 $ with norm $\| \cdot \| := \| \cdot \|_{\ell^2} + \| \cdot \|_{\ell^p}$

Clearly $ p \geq 2 $ so it gains sense calculating the $\ell^p $-norm. According to my calculation this norm is equivalent to the $\ell^2 $ norm, in fact given a cauchy sequence w.r.t $\| \cdot \| $ ...
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49 views

A matrix in $SL(2)$ has it's supremum norm and infimum fulfilled by orthogonal vectors

I am having trouble proving the next statement: If $B\in SL(2)$ and $||B||\neq 1$, for $||B||:= \underset{x\neq 0}{\sup}\big\{\frac{||B(x)||}{||x||} \big\} $, where $||\cdot||$ is the euclidian norm, ...
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38 views

let $B(,)$ and $B'(,)$ be inner products over $\mathbb R$. show there is $c \in \mathbb R$ s.t $B(u,u) \leq cB'(u,u)$

As the title says, let $B(,)$ and $B'(,)$ be inner products over $\mathbb R^n$. show there is $c \in \mathbb R^n$ s.t $B(u,u) \leq cB'(u,u)$ for all $u \in \mathbb R^n$. I have been thinking about ...
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1answer
54 views

natural meaning of vector norms

In class we learned how to calculate 1-norm, 2-norm, and infinity norm. The instructor mentioned that each norm has a natural meaning in the context of a given problem. How can I tell which norm to ...
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32 views

Operators and Differential Equations

I have a question about m-dissipative operators. Thus let $T$ be an closed operator with dense domain. Then $(Tx,x)\leq0$ for all $x\in D(T)$ and $\lambda I-T$ surjective for all $\lambda>0$. I ...
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1answer
104 views

Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it ...
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110 views

Minimum L1 norm may not obtain the sparsest solution?

Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ...
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71 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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18 views

Prove that a sequence of linear maps is bounded iff its matrix representation is bounded.

Let $f$ be an endomorphism of a finite dimensional vector space. We consider the following sequence of maps $(f^p)_p$. $M_B(f)$ is the matrix representation of $f$ in the basis $B$ The following ...
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1answer
71 views

minimum trace norm on the set of matrices with fixed diagonal entries

What is the min nuclear norm (sum of singular values) on all $n \times n$ matrices$A$ whose diagonal is fixed. i.e. $diag(A) = v$ Is it true that the diagonal matrix is a minimizer?
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matrix norm derivative with respect a parameter

What is the result of the following expression $\frac{d}{dt}\left( \|A(t)-B(t) \|\right) $, where $\|\cdot \|$ can be for instance the Frobenius norm?
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51 views

Is delta distribution continuous and differentiable with dual space norm?

I know that delta distribution $\delta : \mathcal S (\mathbf R) \to \mathbf C$ is continuous with usual seminorm and here. I am interested in its continuity with dual-space $H^{-1}(\Omega)$ of ...
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26 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
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1answer
94 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
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Must vectors in $\mathbb{R}^n$ have their “tail” at origin?

I was looking the definition for an $n$-sphere centered at origin with radius $r$: $$\mathbb{S}^n = \{v \in \mathbb{R}^{n+1} : ||v|| = r \}$$ Although I understand that the $||v|| = r$ condition ...
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1answer
40 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
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89 views

Induced norm question.

I'd like to show that the induced 1-norm satisfies: $\|A\|_1=\max_{1 \le j \le n}\sum_{i=1}^n |a_{i,j}|$ I'd appreciate some guidance.
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Prove the following are norms.

I'd like to show that the following are norms: $A \in \mathbb{R}^{n\times n}$ is invertible, so $\|{\cdot}\|\colon\mathbb{R}^n\to\mathbb{R}$ is thus defined: $\forall x \in\mathbb{R}^n$, ...
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56 views

closed and bounded but not compact set of real-valued bounded functions

I'm trying out a problem I was given and this is the statement: Prove, or disprove, that every bounded and closed subset of the set of real-valued and bounded functions on [0,1] equipped with the sup ...
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1answer
30 views

Why all norms define the same relative interior?(Convex Optimization, Stephan Boyd)

When I was reading 'Convex Optimization', Stephan Boyd, I was stopped by Example 2.2. Before Example 2.2 is started, following definition is coming. If the affine dimension of a set ...
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1answer
72 views

Appropriate way to write this equation of linear norm

Assume a continuous and bounded functions $x : \mathbf R \to \mathbf C$, $x(t) \not= 0$ only if $a \leq t \leq b$ and that $|x(t)| \leq 1$ for all $t$. So the signal is finite. I have mappings: ...
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2answers
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Is there any matrix norm $|| \cdot ||$ such that $||A|| \le ||A||_{\infty} /n$?

Matrix norms are equivalent and can bound each other like some examples on Wikipedia. I was wondering if there is a matrix norm $|| \cdot ||$ that can be upper bounded by $||\cdot ||_{\infty}/n$ ? ...
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How to solve an optimization problem with non-convex Frobenius norm constraint?

The form of my problem is: $$ \min_W \|Y-WX\|_F^2-\|V-WU\|_F^2 $$ $$ s.t. \|W\|_F=1 $$ All five variables are matrices. Since the norm constraint is a non-convex one, I have no idea how to solve this ...
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51 views

Equality of $L_2$ norm on $[0, \infty)$

Statement: For a continuous function $y(t)$ defined on $[0,\infty)$, if $\int_0^{\infty}\frac{y^2(\tau)}{1+y^2(\tau)}d\tau<\infty$, $\int_0^{\infty}y^2(\tau)<\infty$. The statement is true if ...
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1answer
100 views

Prove that the square sum of eigenvalues is no more than the Frobenius norm for a square complex matrix

Prove: $$ \sum_{r=1}^{n} |\lambda_r|^2 \le \sum_{i,j=1}^{n} |a_{ij}|^2 $$ the equality holds if and only if $\boldsymbol{A^H A=AA^H} $ for a square complex matrix $ ...
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1answer
29 views

Quadratic Time-Frequency Representation with L2 norm

I have been reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use different norm for different problems in their automatic ECG detection ...
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Contraction map in extending domain from Dense subset to $L^{2}$

This thread is about extending a dense domain $D \subset L^{2}$ into $L^{2}$. I do not understand what Deyton means in his comment about getting contraction map when doing this. I cannot see any ...
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How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
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1answer
38 views

A norm inequality $(AB+BA') \leq |A+A'|*B \leq 2|A|*B$

Suppose that A and B are two $n \times n$ matrices. If $AB+BA'$ and $B$ are both positive definite symmetric matrices, is it true to conclude that $AB+BA' \leq $|A+A'|*B$ \leq 2|A|*B$? $A<B$ we ...
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1answer
27 views

Convex set, vector norms

I'm trying to solve the following question but I'm stuck. "Which of the following constraints define a convex set: ∥x∥ ≤ 1, ∥x∥ = 1, ∥x∥ ≥ 1?" The way to check for convexity is basically the same in ...
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73 views

Contraction mapping and $L^{2}(\Bbb R \times \Bbb R)$ spaces in inequality

I found this: \begin{equation} \lVert Wx \rVert^{2}_{L^{2}( \mathbb{R} \times \mathbb{R} )} \leq \lVert x \rVert^{2}_{L^{2}(\mathbb{R} )}, \end{equation} which I think ...
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40 views

Need help to understand norms

I have some problems to understand norms. At university, my task was to solve the following question: $$ Is\;in\;\mathbb{R}^n (n\ge2)\;a\;norm\;defined\;by\;the\;following\;equations?\\ ...
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45 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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88 views

Prove that the given distance function is a norm

Let the vector space $X=K^3$. For $x=(\alpha_{1}, \alpha_{2}, \alpha_{3}) \in X$, we define $||x||= [(|\alpha_{1}|^2+|\alpha_{2}|^3)^\frac{3}{2} + |\alpha_{3}|^3]^\frac{1}{3}$ Proof that $||·||$ is ...
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75 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
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1answer
47 views

Vector minus its orthogonal projection is the orthogonal projection on the complement?

Assume v is a vector of space V, U is a subspace of V .Pr is the orthogonal Projection- on the left side it is on U, and on the right side it's on the orthogonal complement of U. Is it right this ...
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1answer
28 views

The space $\mathcal{D}((0,T);V)$ and its norm/embeddings?

Let $V$ be a Hilbert space. Define $\mathcal{D}((0,T);V)$ to be the set of functions $u:(0,T) \to V$ such $u$ is compactly supported on $(0,T)$ and is a $C^\infty$ test function. What is the norm ...