# 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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### Propagation of Errors, Variance of a $N$ dimensional vector

I am working on a calculation and I'm trying to see if the last step is true. I tried to simplify all the details of the problem. Let $\mathbf{W}$ be a $N\times N$ matrix, $a=\{a_{1,}a_{2},a_{3}\}$ ...
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### Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
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### Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
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### Is $(1,0,…,0,-n^2,0,0,…) \in \ell^2$?

I am a bit unsure of this because if $n$ is very large then the sum would not be finite but then again the term after the nth term is $0$ onwards.
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### Norm of a matrix formed using a unitary matrix

Suppose, $A$ is a unitary matrix in $M_n(\mathbb{C})$ given by $(a_{i,j})_{1\le i,j\le n}$ which has the property that, for all the basis elements $e_i$, $Ae_i\ne \lambda e_j$ for all $i,j$ ...
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### definition of norm and $\ell_2$ norm

This is from my textbook: I am pretty sure when we have a norm function for a vector, for example $v=(3,4)$, so $\|v\| = 5$, which is the euclidean distance, so why the '2' is deleted rather than ...
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### Show $F_1$ is a continuous linear functional

Let $(X,||\cdot ||)=(C[0,1],||\cdot ||_{\infty})$ and $F_1 :X \rightarrow \mathbb R$ be defined by $$F_1 (f)=\int _{1/2}^{3/4} f(t) dt$$ Show $F_1$ is a continuous linear functional. So we need to ...
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### Norm of a functional is a norm on a v.s. $X^{*}$

Prove that $\| \cdot \|_{X^{*}}$ is indeed a norm on $X^{*}$, the space of bounded linear functionals on a normed space $(X, \| \cdot \| )$. I am not sure what to do in this. I do know that we ...
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### Linear functional: continuous at $x_0=0 \iff$ continuous at all $x$

Let $f$ be a linear functional on a normed space $(X, ||\cdot ||)$. Prove that $f$ is continuous $\iff$ it is continuous at all $x \in X$. Backward direction is trivial: Since $f$ is continuous at ...
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### Relationship between matrix structure and size of matrix norms

I am attempting to compute a smallest norm of a matrix (it can be any norm). Are there any results that allow one to say which norms are smaller for certain classes of matrices (for example, diagonal,...
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### How to show this inequality with matrix norms?

Let $A$ be invertible and $M$ of the same size as $A$. Show that if $$\frac{\|M\|}{\|A^{-1}\|}\le c \le \frac{1}{2}$$ then $A+M$ is invertible and $$\frac{\|(A+M)^{-1} - A^{-1} \|}{\|A^{-1}\|}\le 2c.$$...