0
votes
0answers
10 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
0
votes
0answers
28 views

Custom Norm Function Proof $\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} \right | $

For Vector Space X consisting of ordered pairs of Complex numbers, Can we define the Norm stated below from inner product, in X ? $$\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} ...
1
vote
0answers
23 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
1
vote
2answers
64 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
1
vote
0answers
60 views

Norm inequality with wedge product

Anyone could help me to prove this following inequality? $\displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||} $ where $u\wedge v$ is the wedge ...
0
votes
1answer
20 views

multivariable calculus question norm

Given vector space C([a,b],$ \mathbb{R} $) of continuous functions of [a,b] in $ \mathbb{R}. $ Prove that the function $ \left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt $ is a norm. Also ...
0
votes
2answers
51 views

Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
0
votes
1answer
26 views

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.
0
votes
0answers
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 ...
0
votes
2answers
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 ...
0
votes
1answer
61 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 ...
0
votes
3answers
43 views

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 ...
0
votes
1answer
44 views

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, let $W$ = { f ∈ $C(\mathbb{R})$ | $∫_{-∞}^∞$ | f(x) | $dx$ < ∞} where the ...
0
votes
2answers
46 views

Infinite number of induced norms

While proving that a norm had to come necessarily from a scalar product I have started to wonder about the concept and uniqueness of induced norm. My teacher hasn't clarified to me this doubt, saying ...
6
votes
3answers
101 views

What are some uses for other norms on $\mathbb{R}^n$

We all know and love the standard $1,2,$ and $\infty$-norms on $\mathbb{R}^n$. However, I have never seen anyone mention uses for any of the other $k$-norms that I'm defining as ...
0
votes
1answer
68 views

Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
2
votes
2answers
67 views

Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
3
votes
1answer
63 views

Showing that the zero vector has norm zero

I need to show that this is a property of a norm. I know this is supposed to be straightforward but I am somehow not seeing it. The property is $$\lVert 0\rVert = 0$$ I was trying to use the fact ...
2
votes
3answers
362 views

How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?

I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
1
vote
2answers
248 views

What is the role of supremum in operator norm

An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
1
vote
2answers
67 views

Linear functional $\mathscr{L}(E,F)$

Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$. Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question: How to prove ...
1
vote
1answer
109 views

On norm selection for the solution of an overdetermined linear system

I am considering the following linear system: $Ax = b$ Where: $A$ is $9000 \times 139$ $x$ is $139 \times 1$ and sparse $b$ is $9000 \times 1$ Most of the resources I have found online point to ...
4
votes
2answers
130 views

Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector. What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$? In addition if $\Vert x \Vert_2=1$, what ...
1
vote
0answers
43 views

A modular which is not a metrizing modular (hence not an F-norm)?

I'm taking the terminology from Rolewicz's 1985 Metric Linear Spaces. Given a complex vector space $X$, a modular $m$ is any function $m:X\to[0,+\infty]$ satisfying the following for all $x,y\in X$ ...
3
votes
1answer
47 views

bound on $l_2$ error in approximating a vector with its $t$-sparse representation

How do I prove that for any vector $y\in \mathbb{R}^n$, and any positive integer $t$, \begin{equation} ||y-y_t||_2\:\leq\: \frac{1}{2\sqrt{t}}||y||_1 \end{equation} where $y_t\in\mathbb{R}^n$ is the ...
0
votes
1answer
551 views

compute the similarity between two vectors

Euclidean distance is a measure that may be used to compute the similarity between two vectors. Given a query $q$ and documents $d_1, \ldots, d_n$, we may rank the documents $\mathcal{D} = ...
0
votes
2answers
60 views

Norm on vector space.

I was given a true or false questionnaire to study for my final and do not know if I am right or wrong about these statements. I marked the following statement as True: If $\|\cdot\|$ is a norm on ...
1
vote
2answers
70 views

Operator norm converging to 0 for certain condition

Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
0
votes
1answer
153 views

Minimum of Frobenius norm

I'm not sure if this solvable. I would like to find an orthogonal matrix that minimizes the following Frobenius norm: $$\min||BQ||^2 \\ \text{s.t. } Q^TQ=diag(\alpha_1,...,\alpha_n)$$ where ...
0
votes
1answer
55 views

Norm to evaluate precision

I have two vectors x and y with their respective n coordinates/components and this norm between them: $$ norm = \sqrt{\frac{\displaystyle\sum_{i=1}^{n} (x_{i}-y_{i})^2}{\displaystyle\sum_{i=1}^{n} ...
3
votes
1answer
449 views

How to show the unit ball of the dual norm is also polytope?

Assuming the norm's unit ball is a convex polytope. How can one show that the unit ball of the dual's norm is convex polytope and/or polytope ?
3
votes
2answers
221 views

Matrix proof using norms

I have a linear algebra question I need help with. Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
2
votes
1answer
56 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
1
vote
1answer
72 views

Norm of Polynom

I'm trying to understand the following equation: $$\langle p,q\rangle_p=\int_{-1}^1p(x)q(x)dx\\ \text{Basis: }\{p_1,p_2\}\\ p_1:=2x,\quad p_2:=x-1$$ Norm: $$p_1:q_1=\frac{p_1}{\|p_1\|_p}$$ ...
2
votes
2answers
52 views

Why the max can be found only with normalized vectors?

Can someone explain why the term in the first {} equals the second term in the {} :
1
vote
1answer
78 views

How to show that the scalar product on a vector space extends by continuity to a scalar product on the completion of the vector space?

I'm trying to solve the following problem: Assume $H_0$ is a vector space equipped with a scalar product. Complete $H_0$ with respect to the norm $\Vert x \Vert = \langle x,x \rangle^{1/2}$. We ...
1
vote
5answers
184 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
1
vote
3answers
89 views

Norm in $L^2$ space

I have a brief question regarding the norm of the $L^2$ space defined on an interval $[a,b]$. On various websites I have seen this defined as: $$\|f(x)\| = \int_{a}^{b} f(x)^2 dx$$ However, ...
0
votes
1answer
82 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
0
votes
1answer
104 views

Is there a linear function that is *not* continuous between two normed vector space?

The textbook says that this function has to be continuous at least in the origin for it to be continuous everywhere. But how is it possible that a function is already linear but somehow not ...
1
vote
0answers
117 views

constrained optimization of dot product

Given a real matrix $A$ find a positive vector $x$ of unit length ($x^T x = 1$) for which $x^T A^T A x$ is minimal (closest to $0$). A has size about $1000 \times 20$ and can be written as $[ A_P | ...
6
votes
1answer
548 views

What matrices preserve the $L_1$ norm for positive, unit norm vectors?

It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? ...
1
vote
1answer
91 views

What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
2
votes
2answers
3k views

how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?

In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof: $$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$$ ...
7
votes
1answer
492 views

Does convexity of a 'norm' imply the triangle inequality?

Given a vector space $V$ (for convenience, defined over $\mathbb{r}$), we call $d:V\rightarrow\mathbb{R}$ a norm for $V$ if $\forall \mathbf{u}, \mathbf{v} \in V$ and $\forall r \in \mathbb{R}$ we ...
18
votes
2answers
4k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
0
votes
1answer
151 views

parametrize hypersphere

I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$. Is there a general parametrization of $p$-norm hyperspheres ...
4
votes
1answer
297 views

Prove a basic fact on a linear combination of vectors

Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact? Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i ...
9
votes
2answers
993 views

How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?

I want to prove the following theorem (no idea whether it has a name): Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...