# 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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### 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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### Matrix Norm set #2

As a complement of the question Matrix Norm set and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions: (3) ...
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### 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 ...
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Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm $|... 1answer 95 views ### How can I give a bound on the$L^2$norm of this function? I came across this question in an old qualifying exam, but I am stumped on how to approach it: For$f\in L^p((1,\infty), m)$($m$is the Lebesgue measure),$2<p<4\$, let $$(Vf)(x) = \frac{1}... 1answer 1k views ### Proof of Clarkson's Inequality Trying to find a proof for Clarkson's inequality, which states that if 2 \leq p < \infty, then for any f, g \in L^p, we have that$$\left|\left|f+g\right|\right|_p^p + \left|\left|f-g\right|\...
I have the following matrix norm: $$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$ I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...