Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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How are H-infinity norm and L-1 norm related?

I have trouble understanding the following expressions which I recently encountered in one paper. Suppose that you have the following transfer function: $G(s) = \frac{E_i(s)}{E_{i-1}(s)}, \quad E_i(s) ...
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Extending the closed unit ball in an open set

In a normed space (over the reals), let $C_1 = \{x : \|x\| \leq 1\}$ the closed unit ball, and let $U$ be an open set including $C_1$. In $\mathbf{R}^p$, it is intuitively apparent that you can "...
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Norm of Operator : $Af(x)=f'(x)$

Problem Let : $$A~~:~C^{1}\left([0,1],||.||\right)\to~C\left([0,1],||.||_{\infty}\right)$$ $$Af(x)=f'(x)$$ Where : $||f||_{C^{1}}=||f||:=||f||_{\infty}+||f'||_{\infty}$ I was prove that $A$ operator ...
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Lower bound for the inverse matrix function

Let $X$ be a compact metric space and $A :X \to GL(d, \mathbb{R})$ be a continuous function over a homemorphism $T: X \to X.$ We denote $A^{n}(x)= A(T^{n-1}(x)) \ldots A(x).$ Assume that there is $C&...
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Prove $\|Tx\|_\infty \le \|x\| $ for all $x$ belonging to a normed space

Here is the question: And here is the given answer: I would like to ask in the first line of the proof, why $\|x^*_k\|$ is less or equal than $1$? I guess zero belongs to $B(x^*_k,r)$ for any $k$, ...
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How to proof Lip-constant weakens when size of function class approaches to infinity?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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How to scale a fat matrix to ensure its rows are orthonormal?

I have the following matrix $${\bf B} = \begin{bmatrix} 0&-4.2423&4.2423&1.4871\\ 1.6532&-1.2735& -1.2735&0.0024\\ 0 & -0.2805 & 0.2805 & -0.8823 \end{bmatrix}$$ I ...
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Let $X$ be a Banach Space.Prove that, $X$ is strictly convex iff every points of $S(X)$ is exposed points of $B(X)$.

Some definitions- $X$ is said to be strictly convex or rotund if for all $x,y\in S(X),\ x\ne y$ we have $\left\lVert\frac{x+y}{2}\right\rVert<1$ A point $x_0\in B(X)$ is said to be a exposed point ...
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$X$ is strictly convex (rotund) iff for all $x,y\in S(X)$ with $x\ne y$ we have $\lVert x+y\rVert <2$

A Banach Space $X$ is said to be strictly convex or rotund if for all $x,y\in X$ we have $\lVert x+y\rVert<\lVert x\rVert+\lVert y\rVert$ unless $x,y$ are multiple of each other. We have to prove ...
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How can you proove that every bounded function in $L^1[0;1]$ can be approximated by continuous function in $C[0;1]$?

Here my question, is this true that: Every bounded function in $L^1[0;1]$ can be approximated by continuous functions in $C[0;1]$ It seems to me true as we know that $C[0;1]$ is dense in $L^1[0;1]$, ...
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Is there a rigorous proof that $L_1$ regularization produces sparse results

I'm looking for a rigorous proof that the $L_1$ regularization really produces sparsity, there are multiple intuitive ones like these which I understand but all of these feel like they just explain ...
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How would I prove this inequality regarding positive definite matrices.

Let, $C,A,B \in \mathbb{C}^{n \times N}$, and $C$ and $A$ be hermetian. Suppose that, $$C - B^{*}A^{-1}B,$$ is hermetian positive definite. How would we prove that, $$||C||_{2} \ge ||B^{*}A^{-1}B||_{2}...
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Example of discontinuous convex l.s.c. function on an open convex subset of an incomplete normed space

I'm reading Proposition 0.7. in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is continuous on $C$. (b) If ...
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Does my proof generalize this exercise about continuity of convex lower semi-continuous function?

I'm trying to solve below question (Proposition 0.7.) in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is ...
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Let $C$ be a finite-dimensional convex subset of a normed space. Then the convex function $f: C \to\mathbb{R}$ is continuous on $\operatorname{ri}(C)$

I'm trying to solve below question (Corollary 0.5.) in this lecture note. Let $C$ be a finite-dimensional convex set in a normed vector space $X$. Then every convex function $f: C \rightarrow \mathbb{...
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Proof: $S$ a subset of $l^2(\mathbb{N})$ is a closed subset

I am doing exercice on a book and sometimes or i haven't the solution to the question or i didn't understand their solution. Question: Proove that the subset $S$ that countain all of the sequences of ...
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There exists a ball inside the convex hull of the union of $2$ other balls

I'm trying to show that given a point in the line segment connecting centers of $2$ given balls, there is a ball centered at that point inside the convex hull of the union of that $2$ other balls. ...
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In a normed vector space, the convex function $f:C \to \mathbb R$ is locally Lipschitz if and only if $f$ is upper bounded on an open subset of $C$

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $(X, \| \cdot\|)$ be a normed ...
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Show that $\|v\|=\sup \{|\langle v,w \rangle|, \|w\|\leq 1\}$

I need some help. Let $(H,\langle\cdot,\cdot\rangle)$ a Hilbert space, and for every $v\in H$, defines a continuous function $\langle v,\cdot \rangle:H\to \mathbb{R}$. Show that for every $v\in H$: $$\...
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Computing a norm of a vector, induced by an inner product [closed]

Compute the norm of $\begin{bmatrix}4\\3\end{bmatrix}$, induced by the following inner product $$\langle x\mid y\rangle = 3x_1y_1 - x_1y_2 - x_2y_1 + 5x_2y_2$$ $$\begin{bmatrix}3 & -1\\-1 & 5\...
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Bounded from above convex function on a normed vector space is locally Lipschitz

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) I'm trying to generalize this result to ...
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How many points does a line intersect a sphere in an infinite-dimensional normed vector space?

Let $(E, |\cdot|)$ be a n.v.s. We fix $r>0$ and $x,y \in B(0, r)$ such that $x\neq y$. Here $B(0, r)$ is the open ball centered at the origin and having radius $r$. The set of all points in the ...
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Is $A\subset X$ bounded when $\{\Delta(x):x\in A\}$ is bounded for all $\Delta\in X^*$, $X$ is a normed space and $A\subset X$? [closed]

Let $X$ a normed space and $A\subset X$. Suppose that for all $\Delta\in X^*$, $\{\Delta(x):x\in A\}$ is bounded. Prove that A is bounded
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How to demonstrate this inequality regarding the Rayleigh quotient is true

Let, $\phi(x) = \frac{x^* M x}{x^* x}$ where $x \in \mathbb{C}^{n}$ and $M \in \mathbb{C}^{n \times n}$. How would one demonstrate that: $$|| Mx - \phi(x) x||_{2} \le ||Mx - \kappa x||_{2},$$ for any $...
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If two functions are close apart can I prove the difference of their empirical loss is also small?

I am trying to understand the proof of Theorem 3 in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Basically there exist atleast one $w_{L,e}$ in $\...
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3 votes
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Is there a equivalent norm on $L^p$ induced by inner product?

Suppose $L^p[a,b]$ is the normed space with the usual norm $\|f\|_p=(\int_a^b|f(x)|^p\mathrm{d}x)^{1/p}$. By the parallelogram equality, we know the norm is induced by an inner product iff $p=2$. ...
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Is $\Lambda:X\longrightarrow\mathbb K$ continuous if $\{\Lambda(x_k)\}_{k\in\mathbb N}$ is bounded when $x_k \longrightarrow0$?

Let $X$ a normed space over $\mathbb K$ ($=\mathbb R$ or $\mathbb C$) and $\Lambda:X\longrightarrow\mathbb K$ a linear aplication. Prove that if $\{\Lambda(x_k)\}_{k\in\mathbb N}$ is bounded $\forall(...
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Proof: $C^\infty[0,1]$ is dense in $L^1[0,1]$

Intro: I would like to know if my demonstration of $C^\infty[0;1]$ is dense in $L^1[0,1]$ is correct because I didn't find any complete demonstration of that statement. -(i) As we know from here all ...
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3 votes
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$0 \leqslant a \leqslant b \Rightarrow \|a\| \leqslant \|b\|$ in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a,b\in A$. Therefore $0\leqslant a \leqslant b \Rightarrow \|a\|\leqslant \|b\|$. I'm trying to prove this claim, but apparently it's necessary to use some spectral ...
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4 votes
1 answer
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Proof: All functions in $L^2[0,1]$ are in $L^1[0,1]$

I would like to know if my demonstration of all functions in $L^2[0,1]$ are in $L^1[0,1]$? $\forall f\in L^2[0,1] $ we can split $f$ in two differents spaces: $A=\left \{0\leq x\leq 1:|f(x)|>1) \...
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Let $\{\Lambda(x_k)\}_{k\geq 1}$ be bounded seq. for each sequence $(x_k)$ contained in a normed space, converging to $0$. Is $\Lambda$ continuous?

Let $X$ be a normed space and let $\Lambda:X\to\mathbb{R}$ be a linear function. Out hypothesis are that $\{\Lambda(x_k)\}_{k\geq1}$ is bounded for each sequence $(x_k)\subseteq X$ such that $x_k\to 0$...
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How to prove that $A$ non empty clopen subset of $\Bbb{R^n}$ implies $A=\Bbb{R^n}$?

I want to prove that the euclidean space $\Bbb{R^n}$ is connected. For $x,y\in \Bbb{R^n} $ , $f: [0,1]\to X $ defined by $f(t)=(1-t)x+ty$ is a path.Hence $\Bbb{R^n}$ is path connected and then ...
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What does it mean : infinite norm over the distance between two function? [duplicate]

I am trying to understand the assumption proof of Theorem 3 in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of ...
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Proving $||f||^2_{L^2(\mathbb R^2)}\le 10||f||_{L^1(\mathbb R^2)}||\nabla f||_{L^2(\mathbb R^2)}$ with $f\in C^1,L^1\cap L^2$ and $\nabla f\in L^2$.

Prove that there exists a universal constant $K<10$, for all $C^1$ function $f : \mathbb R^2 \rightarrow\mathbb R$, if $f \in L^1 (\mathbb R^2)\cap L^2(\mathbb R^2)$ and $|\nabla f| \in L^2(\mathbb ...
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Relation between uniform and operator norm

Let $f:U\subset \mathbb{C}\to \mathbb{R}^n$ be a $\mathcal{C}^1$ function. I would like to know the relation between the uniform norm and the operator norm of the differential. For this question to ...
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How do I normalize a vector such that the sum of its squared elements is some arbitrary c?

I am trying to generate points (vectors) from the $L^2$ unit norm hypersphere uniformly at random. This post says to: Generate a random Gaussian $d$-dimensional vector $v$. Generate a random uniform ...
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The topology induced by $p$-product norm

I am reading this answer in which the author said There are many ways to equip $X \times Y$ with a norm, the most natural ones are $\|(x,y)\| = \max{\{\|x\|,\|y\|\}}$ and $\|(x,y)\| = \|x\| + \|y\|$ ...
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What is $\langle x, \varphi \rangle \overline{\langle x, \varphi \rangle}$ equal to?

I'm working on the following question: Let $X$ be a Hilbert space with dim($X$)≥2 and let $T\in B(X)$ be of the form $$Tx=\langle x,\varphi\rangle\psi$$ Show that $||T||=||\varphi|| \hspace{0.3em}||\...
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Comparison of $L^p$ and $L^q$ norms to establish the inclusion between corresponding spaces

We can deduce that; for any $x \in \ell^p,$ the space of $p$-summable real sequences ($p \geq 1$), $$\lVert x \lVert_q \leq \lVert x \lVert_p,~p \leq q < \infty,$$ by just letting $e=\frac{x}{\...
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The norm in the n-simplex

İ have this question and my attempt to solve it as the following : Consider an n-simplex $[p_0, p_1,...,p_n]$. If $a,b \in [p_0, > p_1,...,p_n]$, then show that $||a − b|| \leq \sup_i ||a − p_i||$....
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What's the relation between (strict) convexity of unit balls and shortest distance paths in $l_p$ metric?

I'm reading the book Geometry of Quantum States by Bengtsson and Zyczkowski. They have a brief discussion on $l_p$ norms. Depending on circumstances, different choices of p may be particularly ...
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Tricky $L^p$ Spaces Inclusions [duplicate]

In sampling theory, $L^p$ spaces are very important mathematical constructs. To this, we are currently studying some of their properties in class. In this regard, one exercise that I just cannot wrap ...
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Schauder basis and convex combinations

Let us suppose that $(X, \lVert \cdot \rVert)$ is a normed space over $\mathbb{R}$ which has a Schauder basis, that is, there is a sequence of vectors $(x_n)_n$ in $X$ such that for all $x \in X$ ...
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Proving two norms are not equivalent. Counterexamples. $V = C([0,1])$, $||f||_{\infty} = max |f(x)|$, $||f||_{*} = max |xf(x)|$.

$V = C([0,1])$, $||f||_{\infty} = max |f(x)|$, $||f||_{*} = max |xf(x)|$. The question asks: if the two norms are equivalent? And if not, why? How do I deduce whether the two norms are equivalent or ...
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Compute sup norm of sequence of functions.

Consider the operator $T: C^2[0,1] \subset C^1[0,1] \to C^1[0,1]$ defined by $Tf=f'+f''$. Compute $\| T e^{-nx } \|_{\infty }$ and $\| T x^n \|_{\infty }$. My attempt. First I tried to compute $\| e^{...
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4 votes
1 answer
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Practical uses of $p$ norms for $p\notin \{1,2,\infty\}$?

We all love normed spaces, but it seems like the $1$-, $2$-, and $\infty$-norms get the lion's share of the love. That's not admittedly not without good reason, but what of the other unsung norms with ...
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3 votes
1 answer
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The space of Lipschitz continuous functions is dense in that of uniformly continuous functions?

Let $(X,d)$ be a metric space. Then The space of bounded uniformly continuous functions is dense in that of bounded continuous functions w.r.t. the supremum norm. ref The space of bounded Lipschitz ...
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For $p\ge 1$, $x\ge 0$ and $y\ge 0$, $|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$?

I vaguely remember seeing an inequality of the form: $$|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$$ for $p\ge 1$, $x\ge 0$ and $y\ge 0$. Is this correct? If so, how is it proven? Can it be ...
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Derivative of l2-norm w.r.t matrix

I have a matrix A which is size of mm, a vector b which is size of m1. I'd like to derivative of the following function f(A) w.r.t A: $$f(A)=||A*b||_2$$ I need to find the first order Taylor expansion ...
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8 votes
2 answers
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Exercise 6.A.17 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. I am worried if my solution is ok.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 6.A.17 Prove or disprove: there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by $$||(x,...
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