# Questions tagged [norm]

This tag is for the questions relating to the norm which is a function on a vector space $X$ that generalizes notion of length of vector in general vector spaces. A vector space $~X~$ with a distinguished norm is called a normed space.

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### Will $\sqrt{h \sum_{i =0}^{N-1} （1 - u_i^2)^2} < C_1$ imply there exist $C_2$ satisfies $\sqrt{h \sum_{i=0}^{N-1} u_i^2} < C_2$ [closed]

Assume the interval $[a, b]$ is divided by uniform grids $x_i = a + i * h, i = 0, 1, \cdots, N$, where $h = \frac{b - a}{N}$, $\mathbf u = [u_0, \cdots, u_{N-1}]^T$ is a grid funciton, will the ...
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### Specific matrix norm

I have a real, limited scalar field $S (x, y)$ which I describe with a matrix $M$ with an approximate value of $S (x, y)$ within each cell. The bigger the matrix, the more precise this description ...
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### Why does the metric on a Schwartz space generate the same topology as the family of seminorms?

I am reading Rauch's "Partial Differential Equations", and he makes a jump I don't understand. He defines the Schwarz space as the space of $C^\infty$ functions that decrease faster than any ...
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### Householder transformation matrix proof norm [closed]

Definition: Let $x \in \mathbb R^n$ and $Q:=(I-2*ww^T/w^Tw)$ the Householder matrix Exercise: Is the vector $w = x + ∥x∥e_1$ not equal to zero, than is $Qx = −∥x∥e_1$ Is $w = x − ∥x∥e_1$ not equal to ...
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### Derivation of within point scatter $W(C)$

I'm reading the book "The Elements of statistical learning". In the section about K-means clustering they derive an equation regarding the "within point scatter" which is a ...
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### Cauchy Schwarz manipulation

I am trying to manipulate the upper bound below using Cauchy Schwarz, so that the norm of the vector a, $||a||$, appears on the right hand side ($a_i$ itself is a scalar, $a$ is the vector). Which I ...
Let $V$ be an inner product space of finite dimension. Let $v_1, v_2, ... , v_m$ be orthonormal vectors in $V$ and $W=\operatorname{sp}\{v_1, v_2, ... , v_m\}$. let $v$ be some vector and $\alpha _i=\... 1answer 17 views ### norm and projections on inner product space How do I show that if$\Vert Px-Qx \Vert <\Vert x \Vert$for any$x\in V$not$0$, then$\dim\left(M\right)=\dim\left(N\right)$.$V$is an inner product space and$M, N$are sub-spaces of$V$.$P$... 1answer 24 views ### Total variation norm of probability measures related to$L_1\$-norm?
On Wikipedia the following is stated: I don't see how, if the set is countable, $$\delta (P, Q) = \sup_{A \in \mathcal{F}} \vert P(A) - Q(A)\vert = \frac{1}{2} \vert \vert P- Q\vert \vert _1$$ holds....