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

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Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
0
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0answers
20 views

Linear Algebra L2 minimization

Im really confused about how to solve this question or even what its asking. Any help would be much appreciated! Let A be an m x n real matrix ($m \gt n$). Let x* be the minimizer of $||Ax - b||^2 + ...
0
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1answer
39 views

Proof: $ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$

I am looking for a proof of the following:$$ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$$ For positive hermitian matrices A and B, and a unitarily invariant norm $ |||\cdot|||$.
0
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0answers
18 views

Proof of dual norm relation: $\frac{1}{q} + \frac{1}{r} = 1$

Recall: $\|\|$ is a norm in $R^n$, and its dual norm is defined as $\|z\|^*=\text{sup}_{\|x\|\leq1}z^Tx$. If $q$-norm and $r$-norm are dual norm, then we have the following relation: ...
1
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2answers
28 views

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 ...
-1
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0answers
30 views

Set of continuous functions that vanish at infinity is complete

Why is it easy to see that a set of all continuous functions $C_0$ that vanish at infinity implies that each $f\in C_0$ is bounded and the set is complete with respect to the uniform (sup) -norm? ...
3
votes
5answers
49 views

Equivalence of norms: $\|x\|$ and $\|x\|_1=\|x\|+\lvert\,f(x)\rvert$

Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous. I've ...
1
vote
1answer
25 views

Is $\left\|A^TA(x-y)\right\| = \left\|A^TA\right\|\times \left\|x-y\right\|$ correct? $A \in \mathbb{R}^{n \times n}$

In the derivation of following, I meet a dumb problem: Note: 1. $\left\|\: \cdot \,\right\|$ is the $l_2$ norm. 2. $A \in \mathbb{R}^{n \times n} $ 3. $x,y \in \mathbb{R}^{n}$ ...
1
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0answers
20 views

finding Interior Points of a set

In the normed space $(\mathbb{R}^2, ||(x_1,x_2)||:=|x_1|+|x_2|)$ I want to find Interior Points of $$ \{ (x,1/n) ~~\big|~~ x\in \mathbb{R} \text{ and } n\in \mathbb{N} \}. $$ I guess that the ...
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0answers
27 views

What is the norm for the product of normed spaces?

Suppose $(\Omega, \Sigma ,\lambda)$ is a probability space and $X_i=L^2(\Omega,\Sigma,\lambda ,[0,1])$ with norm $||.||_{L^2}$ for all $i\in I$, $I$ is finite. Is there any natural norm for the ...
2
votes
1answer
70 views

Proof: $|||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| $?

I am looking for a proof of the following: \begin{equation*} |||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| \end{equation*} Where A, B are positive, hermitian matrices, and $|||⋅|||$ ...
1
vote
1answer
20 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...
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4answers
39 views

Cauchy-Schwarz, how does it work in this example?

Consider $\|\cdot\|_2$ such that $\|x\|_2 = \left(\sum_{i=1}^n |x_i|^2\right)^{1/2}$. Let $A \in \mathbb{R}^{n\times n}, x\in \mathbb{R}^n$, then $$\begin{align} \|Ax\|_2^2 & = \sum_{i=1}^n ...
2
votes
0answers
36 views

How to prove that $\|x+y\|_p<\|x\|_p+\|y\|_p$? [duplicate]

Let $l_p=\{(x_n)\in\mathbb{R}^\mathbb{N}: \sum_{n=0}^\infty |x_n|^p<\infty\}$ and consider the following norm in $l_p$: $$\|x\|_p=\left(\sum_{n=0}^\infty|x_n|^p\right)^{1/p}$$ for ...
1
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2answers
28 views

Norm is sup of inner products (proof).

Let $V$ be a vector space with an inner product $\langle.,. \rangle$ and associated norm $|| . ||$ Then: Could I have a proof of this fact?
6
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2answers
5k views

showing that l2 norm is smaller than l1

How can I show that L2<=L1 $||x||_1\ge ||x||_2$ and also we have that $\|x\|_2\leq \sqrt m\|x\|_{\infty}$ regarding the first part, can I say that: $$ ...
0
votes
1answer
22 views

On the proof of the continuity of the inner product.

I am having problems with the following proof and I need to fill in some details: I understand that continuity is being proven by the sequence definition but I do not get why (a) follows ...
1
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2answers
29 views

Show that properties of norm are satisfied

Show that \begin{align} & \|y\|_M= \max_{a \leq x \leq b} |y(x)| \tag 1 \\[8pt] & \|y\|_1=\int_a^b |y(x)|\, dx \tag 2 \end{align} satify the properties of a norm in $C[a,b]$. That's what I ...
3
votes
1answer
119 views

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Suppose that $ \star: V \times V \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property? $$ \forall x,y \in V: \quad \| x \star y \| = \| ...
0
votes
1answer
40 views

What exactly is the distance of two elements in $C[0,1]$?

If $C[0,1]$ — the set of all continuous functions from $[0,1] \rightarrow \mathbb R$ — is equipped with the metric $||\cdot||_1$ (1-Norm), then what is the distance between ...
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2answers
30 views

The function is not continuous

$$C([a,b])=\{ f: [a,b] \to \mathbb{R} \text{ continuous} \}$$ $C([a,b])$ is a linear space. For $f \in C([a,b])$ we define $\|f\|_{\infty}:= \sup_{x \in [a,b]} |f(x)|$ and easily it can be shown ...
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1answer
34 views

Hilbert space L2 - inner product

I have a problem with one exercise. I have to prove that $L^2$ space is Hilbertian. So I think that the best way is to check out inner product by definition of norm, so: \begin{equation*} ...
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0answers
27 views

when the spectral radius of a matrix product is equal to the product of spectral radius?

The question is simply as follows, when do we have the following equality? $\rho(AB)=\rho(A)\rho(B)$.
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votes
1answer
26 views

operator norm of a linear transformation, given by the transformation matrix

Consider $\mathbb{K}^n$, $\mathbb{K}^m$, both with the $||.||_1$-norm, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. Let $||T|| = inf\{M ≥ 0: ||T(x)|| ≤ M ||x|| \space \forall x \in ...
1
vote
1answer
18 views

How to prove that $N_g\equiv ||\cdot ||_{\infty}$ iff $g^{-1}(\{0\})=\emptyset$?

Let $E=\mathcal{C}[0,1]$, and $g\in E$, define $N_{g}(f)=||fg||_{\infty}$, the I have to prove that $N_g$ is equivalent to $||\cdot||_{\infty}$ iff $g^{-1}(\{0\})=\emptyset$. The $'\Leftarrow'$ ...
0
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1answer
980 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
0
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1answer
25 views

gradient of hadamard L1 norm

What is the gradient of $\lVert B-A\circ X\lVert_1$ with respect to $X$. $\circ$ is the hadamard product. $A,B$ are constants
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2answers
87 views

I want some help in functional analysis [closed]

I want sone help in functional analysis : $1)$ consider the vector space $X$ of all real -valued functions which are defined on $R$ and have derivatives of all orders everywhere on $R$ define ...
1
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1answer
26 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
3
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2answers
2k views

Gradient of squared Frobenius norm

I'd like to find the gradient of $\frac{1}{2} \big \| X A^T \big \|_F^2$ with respect to $X_{ij}$. Going by the chain rule in the Matrix Cookbook (eqn 126), it's something like $\partial \, ...
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votes
1answer
31 views

Prove norm does not come from inner product.

I know I have to show it does not satisfy the parallelogram law but I don't know how to apply it.
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1answer
55 views

Prove norm doesn't come from inner product.

Please help me prove this. I'm not sure how to apply the parallelogram law to the norm.
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1answer
33 views

Test for normability of a metric on a Banach space

If d is a metric on a (finite dimensional) Banach Space and there exist norms $\|-\|_1$ and $\|-\|_2$ and constants $C_1,C_2 \in [1,\infty)$ satisfying: \begin{equation} C_1\|x-y\|_1 \leq d(x,y) \leq ...
0
votes
1answer
34 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 ...
0
votes
1answer
56 views

Gradient of the elastic net with extra terms

Can anyone tell me the gradient of the below function (w.r.to X) $$ argmin_{X} ~~\frac{\lambda}{2}\lVert X\lVert_2^2 + \lVert X\lVert_1 + tr\bigg(\Delta^T\Big(diag(X)-X\Big)\bigg) + ...
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3answers
50 views

Prove $||\lambda x_1 + (1-\lambda) x_2 - y|| \leq ||x_1 - y||$

Assume we have have $3$ points $x_1, x_2$ and $y$ and $||x_1-y||=||x_2-y||$. How do we prove that the distance between $y$ and the convex combination of $x_1$ and $x_2$ is smaller than that between ...
1
vote
1answer
54 views

norms of row matrices

Let $x_1,\ldots,x_m,y_1,\ldots,y_m$ be $k\times k$ sqaure matrices and assume $\|x_j\|\leqslant\|y_j\|$ for all $j=1,\ldots,m$ (the norm in $B(\ell_2^k)$). Now define the block matrices $x,y\in ...
1
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1answer
37 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 ...
2
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2answers
20 views

norm of product (matrix$\times$vector)

How can i prove that: If $B=AA^T$ then $x^T Bx=\|A^T x\|^2_2$ and $x^T B^{-1}x=\|A^{-1}x\|^2_2$ were x is a vector and A is a $n\times n$ matrix?
3
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2answers
37 views

What is the range of the operator $T$ I mean I want to determine $R(T)$

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
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0answers
20 views

taking a tricky limit $\lim_{p\rightarrow \infty} \int_{\mathbb R^N} \left( \frac{|\nabla u|}{\|\nabla u\|_p}\right)^{p-2} \cdots $

$$\lim_{p\rightarrow \infty} \int_{\mathbb R^N} \left( \frac{|\nabla u|}{\|\nabla u\|_p}\right)^{p-2} \frac{\nabla u}{\|\nabla u\|_p}\cdot \nabla v dx $$ where $|\cdot|$ is the Euclidean (2) norm and ...
0
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4answers
52 views

Difficulty in understanding a step in a definition in the book Walter Rudin

In the book Principles of mathematical analysis by Walter Rudin,He writes: "For $ A\in L(\Bbb R^n,\Bbb R^m)$, define the norm $||A||$ of $A$ to be sup of all numbers $|Ax|$, where $x$ ranges over all ...
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0answers
15 views

Operator norm of symmetric Matrix in Hilbert Space with Hermitian Inner Product

Assume we have a postive definite real matrix $P$ and we define an inner product on a finite dimensional hilbert space $\langle x, y \rangle = x^\top P y$ and clearly the induced norm is $|| x || = ...
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0answers
7 views

controllable system, matrix exponential norm

suppose we have $(A,B)$ be a controllable pair. Can I find a feedback control gain $K$ such that $A_c=A+BK$ is Hurwitz, which also satisfies that $||e^{A_ct}||\leq a e^{(-\lambda t)}$ and ...
0
votes
1answer
40 views

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, ...
1
vote
1answer
22 views

Subset of the sequence space that's closed and bounded but not compact

Consider the sequences space $l^1 = \{a = (a_n)_{n \in \mathbb{N}_0} \subset \mathbb{C}, \sum_{n = 0}^\infty|a_n|< \infty\}$ with the norm $||a||_1 = \sum_{n = 0}^\infty|a_n|$. I want to show that ...
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votes
1answer
19 views

Problem About Norm

Show: If $A$ is a symmetric matrix and $\|A\|_F\leq1$, then prove that $I-A$ is a positive semidefinite matrix.
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1answer
38 views

conditions for norm of linear bounded operator to satisfy $\lvert T_x (y) \rvert = \lVert T_x \rVert$.

Let $x = (x_n)_{n \in \mathbb N} \in l^\infty$ and let $T_x : l^1 \rightarrow \mathbb F$ be defined by $T_x (y) = \sum_{n=1}^\infty x_ny_n$. What condition on $x$ is needed so that there exists $y \in ...
0
votes
1answer
34 views

Upper bound of Frobenius norm of product of matrices.

I'm trying to prove that $||AB||_F\leq||A||_2||B||_F$. As far as I know it isn't a hard problem but I was stuck. Any ideas?
2
votes
0answers
31 views

a norm is symmetric if and only if it is unitarily invariant

how can I prove this : A norm on $\mathcal{M}_n(\mathbb{C})$ is symmetric if and only if it is unitarily invariant ? My attempt I know that a symmetric norm is a norm which verifies : $$N(ABC)\leq ...