# Tagged Questions

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

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### Norms Induced by Inner Products and the Parallelogram Law

Let $V$ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $\lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
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### 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 ...
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### Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
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### 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 ...
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### A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
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### 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 ...
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### What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are ...
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### On the convexity of element-wise norm 1 of the inverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$\|A\|_1= \sum_{i,j} |A_{i,j}|.$$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
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### why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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### From norm to scalar product

In the Eculidean Space, one can automatically define a norm if a specific scalar product is given. On the contrary, it's not always true. A p-norm is a scalar product if and only if p=2. My question ...
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### How does one prove that the spectral norm is less than or equal to the Frobenius norm?

How does one prove that the spectral norm is less than or equal to the Frobenius norm? The given definition for the spectral norm of $A$ is the square root of the largest eigenvalue of $A*A$. I don't ...
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### Is a completion of an algebraically closed field with respect to a norm also algebraically closed?

Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...
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### Counterexamples of Arzèla Ascoli theorem for non-obeyed criteria

I had an exam on functional analysis some time ago, and one of the questions I couldn't make any sense out was the following: Let $\Omega\subset \mathbb{R}$ and $\{f_n\}$ a sequence of continuous ...
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### Difference between orthogonal and orthonormal matrices

Let $Q$ be an $N \times N$ unitary matrix (it's columns are orthonormal). I understand since Q is unitary, it would preserve the norm of any vector $X$, i.e, $||QX||^2=||X||^2$. My confusion comes ...
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Does anyone know a proof of the following problem? Problem: Show that for any three vectors ${\bf x}, {\bf y}, {\bf z}\in \mathbb{R}^d$ the following holds, $$\|{\bf y} - {\bf z}\|\cdot \|{\bf x}\| \... 2answers 174 views ### L^2 Bounds for Markov Chains. Consider a non-negative, square stochastic n \times n matrix P (rows sum to one, P is ergodic). We are interested in characterizing the set of n \times n invertible matrices A such that we ... 1answer 220 views ### When does \|z^2\|=\|z\|^2 Let k \in \mathbb{Z} and consider the field extension K := \mathbb{Q}[\sqrt{k}]. Define a norm on K given by \|p+q\sqrt{k}\| := \sqrt{p^2+q^2}. For any z \in K, I was interested to know when ... 1answer 214 views ### Proving Holder's inequality for Schatten norms Sticking to the finite dimensional case, Holder's inequality for Schatten norms is given by$$\left\|AB\right\|_{S^1}\leq\left\|A\right\|_{S^p}\left\|B\right\|_{S^q}$$for A,B n\times n ... 0answers 524 views ### Monotone matrix norms [Ciarlet 2.2-10] Let \mathscr{S}_n be the set of symmetric matrices and \mathscr{S}_n^+ the subset of non-negative definite symmetric matrices. A matrix norm \|\cdot\| to be monotone if$$A\in\...
I am reading this book where at page 27 following definitions about weighted inner product and weighted norms are given. Let $M$ and $N$ be Hermitian positive definite matrices of order $m$ and $n$...