# 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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### How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
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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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### Maximum subset sum of $d$-dimensional vectors

This is a $d$-dimensional generalisation of the post Inequality with Complex Numbers. (See my comment under Robert Israel's answer.) Generalising Potato's proof for $d$-dimensions, we can show the ...
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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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### 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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### Understanding weighted inner product and weighted norms

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 $n$...