# 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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### Source/explanation for this matrix inequality

Here it is: $$z^\top M^{-1} M^{-1}z \le \|M^{-1}\| z^\top M^{-1} z.$$ Where $\pmb M$ is positive definite symmetric, $z$ is a vector in $\mathbb{R}^p$ (not necessarily normed!) and $||\pmb M||$ is ...
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### Minimizing the Frobenius Norm

I would like to minimize the following expression with respect to matrix X: $$\left \| A-BX \right \|_{F}$$ where A and B matrices are given and all the matrices have positive integer elements. Any ...
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### Show that the sup-norm is not derived from an inner product

I am trying to show that the norm $$\lVert{\cdot} \rVert _{\infty}=\sup_{t \in R}|x(t)|$$ does not come from an inner product (the norm is defined on all bounded and continuous real valued functions)....
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### Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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### Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
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### Prove a basic fact on a linear combination of vectors

Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact? Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i x_i=0$...
I want to show that $$\| x \| = \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{\left| x_n \right|}{1+\left| x_n \right|}$$ is a norm. I'm fine showing positivity and the triangle inequality, to show the ...