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

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when norm of an operator is given by max of eigen values modulas

Could any one tell me how this $\|x\|^2=\|x*x\|$ and the rest of it? I know $\|x\|=\|x^*\|$, I also understand $x^*x$ is hermitian and so diagonalizale but then did not understand the norm square ...
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compactness of $L^2$ normed space

I have no idea how, where to start. I mean that we can show the compactness of the set via existence of convergent subsequence. But how can I take it? Please give a clue. This is my problem Show ...
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26 views

Shortest distance from a point to a a Hyperplane

how could I prove the following using Lagrange optimization? Prove that the shortest distance from the hyperplane $$H= \{\vec{x} \in \mathbb{R}^{n} : \vec{a} \cdot\vec{x}=b\} $$ to a point ...
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Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
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Finding the norm of a linear functional

This is a basic question of functional analysis, but I want to know how to... Find the norm of the linear functional $f$ defined on $C[-1,1]$ by $$f(x)=\int_{-1}^0 x(t) \, dt - \int_0^1 x(t) \, ...
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Subordinate norm is equal to $\Vert A\Vert_M=\Vert MAM^{-1}\Vert$ for the norm $\Vert x\Vert_M=\Vert Mx\Vert$

Let $M\in \mathcal{M}_n(\Bbb{K})$ a nonsingular matrix, we have $\Vert x\Vert_M=\Vert Mx\Vert$ is a norm over $\Bbb{K}^n$. I have to prove that the subordinate norm is equal to $\Vert A\Vert_M=\Vert ...
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Distance of two vectors under L_inf norm

Very simple question: suppose I have two vectors $a = (1,-2)$ and $b = (4,2)$. Under the L_inf norm, would the distance between them be $abs( ||a||_{inf} - ||b||_{inf}) = abs(2 - 4) = 2$? Is this the ...
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Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
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Proving result on matrix norms

How do I prove that, letting, for $A\in\mathbb{C}^{n\times n}$: $$(a)\quad\|A\|_1=\max\limits_{i=1,\cdots,n}\left(\sum\limits_{j=1}^n|a_{ij}|\right),$$ $$(b)\quad\|A\|_2=\rho^{\frac12}(A^HA),$$ with ...
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Model $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty $ into standard form

I need to solve the following convex optimization problem: $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty$ s.t $x ...
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Seminorm proof of a function

I have an example in a book which is not very clear to me : let $E$ vector space made of numerical functions (or complex) $f$ defined on a set $A$. $\forall a \in A, N_a : f \rightarrow |f(a)|$ is a ...
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58 views

Norm of a Vector

Suppose $A\inℝ^{n,n}$. We Define $$ \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|}$$ Show it is a norm. Any thoughts?
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Convexity of Norm of Max

Let $p \ge 1$. Show the convexity of the function $h:\mathbb{R}^k \rightarrow \mathbb{R}$ defined as: $$h(\textbf{z})=\left(\sum\limits_{i=1}^k \max\{z_i,0\}^p \right)^{1/p}$$
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Lengths of curves - Arc length

If $\gamma\colon [a,b] \rightarrow \mathbb{R}^d$ is a differentiable curve we usually define $$ \text{length}(\gamma) := \int_a^b \|\dot{\gamma}(t)\|_2\, \text{d}t,$$ where $\|\cdot\|_2$ is the ...
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62 views

Open set generated by two equivalent norm

Suppose X is an vector space and $||.||_a$ and $||.||_b$ are two norm on it, how i can proof that ''This two norms are equivalent iff they generate same open sets."? P.S.: Sense of question made by ...
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Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $E[||x||_2]$, $x $~$ ...
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What to call the Euclidian norm divided by a constant

I'm using the Euclidian distance $d_{2}$ divided by a constant $T$, i. e. $\frac{d_{2}}{T}$. However, I'm not sure what to call this. I'd like to keep things simple so I thought maybe "scaled ...
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51 views

Induced Matrix Norm

I have trouble following a proof of the induced Norm $||\cdot||_1$ The proof can be found here: ...
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61 views

Norm of the multiplication operator

Let $f \in L^\infty[0,1].$ It is clear that the norm of the multiplication operator $M_f : g \mapsto fg$ on $L^p[0,1]$ is $\|f\|_\infty.$ What happens in the noncommutative situation? Let us ...
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Natural norm for the ring $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $?

I am working on showing that $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $ is an euclidean domain. There was a similar problem showing that $\{a+b\sqrt{-2}$ | $a,b \in \mathbb Z \} $ was an euclidean ...
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Prove of beaing Norm function

Prove that relations beneath have conditions for being a norm function A) for $C^{n}: \left \| x \right \|= \left (\sum_{j=1}^{n}\left | \xi _{j} \right |^{2} \right )^{1/2}$ B) for ...
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Matrix Norm Lemma

There is a lemma claims that : $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|) = ||A|| $ I'd like to know how come $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|)$ because it does not make sense ...
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166 views

Minimize norm - Least Squares - Linear Algebra

Given $Ax = b$, I know how to use least squares to minimize $||Ax-b||^2$. How do I minimize the 2-norm ($||x||_2$) and the Frobenius norm of $x$?
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Restrictions on a Matrix-Vector product

Suppose I have a $m\times n$ matrix $\mathbf M$, and a unit vector $\hat v$, of dimension $n$. What restrictions do I need to apply to $\mathbf M$ so that $\lVert \mathbf M\cdot \hat v\lVert \leq 1$ ...
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170 views

Normed linear space with two norms that are not equivalent, one is complete, what about the other?

I have been searching for an answer to the following question: Given a normed linear space $V$ and two norms that are not equivalent, but $\exists K\in\mathbf{R}$ such that $\|v\|_1\leq K\|v\|_2$ ...
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115 views

Derivatives of Norms and Absolute Values (distributions)

For example we have for $x \in \mathbb{R}$, $$\frac{\partial}{\partial x}\left| x\right| = 2\Theta(x) -1 $$ and thus $$\frac{\partial^2}{\partial x^2}\left| x\right| = 2\delta(x) $$ We also have, ...
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proof for matrix norms

How do I prove these two inequalities on matrix norms: $\Vert A \Vert_1 \leq n\Vert A \Vert_\infty,$ $\Vert A \Vert_1 \leq \sqrt{n}\cdot\Vert A\Vert_F$ , where A is $m$-by-$n$ real matrix.
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Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm?

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm ? I know that the algebra of polynomials is dense in algebra of continuous functions, wrt to sup norm, And I know that if $f ...
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Prove that the operator norm is a norm

Exercise: Prove that the operator norm of the set $S$ of all linear operators $L:R^n\to R^m$ defines a norm on $S$ Definition of norm: A positive function $\| .\|$ on a real vector space $V$ is a ...
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proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
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Example for a two norms and vector space which are equivalent on this Vectorspace

I need an example where the two norms and the vector space are equivalent on the vector space.
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62 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
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Integral of a function is not affected by altering the function values at zero-measure set

I'm studying about Fourier analysis from a book Fourier analysis and its applications, Folland 1992 and I have one point in the source I need clarification about: On page 69 it is stated that: "The ...
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when linear mapping keeps monotonicity of $L_2$ norm

Consider an arbitrary vector $\alpha$ from vector space $R^p$, a linear mapping $A: \alpha\rightarrow A\alpha$ transforms $\alpha$ to $A\alpha$ in space $R^q$. What condition should $A$ satisfy so ...
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max induced norm of matrix

I have to prove that matrix norm $||A||_\infty$ induced by vector norm $||x||_\infty = \smash{\displaystyle\max_{1 \leq k \leq n}} |x_k|$ where $x_k$ is k-th element of vector can be disribed by ...
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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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28 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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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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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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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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83 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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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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39 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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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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Prove that this series converges?

I have a Banach space $X$ and a linear operator $A \in L(X)$. $A$ is bounded such that $||A|| <1$. I then have to show that $$log(I-A)=\sum_{n \ge 1} \frac {A^n}n$$ converges. All I can come up ...
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Norm of inverse, Banach algebra, Proof of Gelfand-Mazur theorem

In a proof of the Gelfand-Mazur theorem, I read that $$ \lim_{\lambda\to\infty}\|(a-\lambda 1)^{-1}\|=0 $$ (where it is assumed for the sake of contradiction that the inverses in the norm exist for ...
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64 views

Norm of a bounded operator

Let $\phi \in C[0,1]$ and $T_{\phi}:C[0,1] \rightarrow \mathbb{R}$ be: $T_{\phi}f = \int_0^1 f(x)\phi(x)dx$ prove that $T_{\phi}$ is a continuous, linear functional and that $||{T_{\phi}}|| = ...
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37 views

When is $c v^\top y y^\top v \ge ||v||^2$

Given $c\in R$ being some constant, $v, y \in R^n$, I want to find conditions for which the following inequality holds true: $$c v^\top y y^\top v \ge ||v||^2$$ EDIT: Note that $y y^\top$ is an $n ...
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40 views

Norm of a vector is determined by evaluation of linear functionals on it: can this be proved without the Hahn-Banach theorem?

Let $V$ be a normed vector space over the field of real numbers, $\mathbb R$, and let $x_0 \in V$ be fixed. I know how to prove $$\|x_0\| = \sup_{f \in V^*, \|f\| = 1} |f(x_0)|$$ using the Hahn-Banach ...
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69 views

Prove that $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a norm for a complete space

Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space. I can't prove the completeness of this ...