# 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 and equivalence classes question

Let $f\in C[0,1]$. Recall two of the norms we considered in class: $$\|f\|_\infty = \sup_{t\in[0,1]}|f(t)|, \quad \|f\|_1 = \int_0^1|f(t)|\ \mathsf dt.$$ Consider the space $C^1([0,1])$ of ...
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### What is the meaning of $||x||=\sqrt{\langle x,x\rangle}$

I understand that a norm assigns a length to each vector in a vector space. I have been told that $$||x||=\sqrt{\langle x,x\rangle}$$ is a norm. So does this equation find the length of vector $x$, ...
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### does $\|A\| > \|B\|$, $A$ and $B$ matrices, imply that $\|Ax\| >\|Bx\|$ for all $x$ in some vector space?

I'm wondering if I can prove the first inequality in a question that I am working on, does that make the second inequality automatic? Thanks,
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### Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
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### Subset of the sequence space that's closed and bounded but not compact

Consider the sequences space $l^1 = \{a = (a_n)_{n \in \mathbb{N}_0} \subset \mathbb{C}, \sum_{n = 0}^\infty|a_n|< \infty\}$ with the norm $||a||_1 = \sum_{n = 0}^\infty|a_n|$. I want to show that ...
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### Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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### When does the $L_1$ convergence imply almost everywhere convergence?

I know that $L_1$ convergence implies existence of an almost everywhere converging subsequence. But I was wondering, can you tell me some extra conditions on functions that make $L_1$ convergence ...
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### Supremum vs. Maximum in the definition of the Lp norm [duplicate]

The $L_p$ norm $||A||_p$ is defined as $$\sup_{x \neq 0} \frac{||Ax||_p}{||x||_p} = \max_{||x||_p = 1} ||Ax||_p \tag{1}$$ I'm not quite getting why the LHS uses $\sup$ but the RHS uses $\max$. I ...
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### find linear functional norm

$C[-1,1]$ above $$f(x)=\int_{-1}^{0}x(t)dt-\int_{0}^{1}x(t)dt$$ What is norm of the f linear fucntional? I tried to solve using definition of norm but I couldn't find result. Please can you give ...
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### Does isomorphism between normed space implies equivalence of norm?

Let $E$ be a vector space, $\|\cdot\|_1,\|\cdot\|_2$ be two norms on $E$, if $T:(E,\|\cdot\|_1)\to (E,\|\cdot\|_2)$ be an isomorphism (linear bijection, $T,T^{-1}$ are bouned), then does this imply ...
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### How to prove that 2-norm of matrix A is <= infinite norm of matrix A

Now a bit of a disclaimer, its been two years since I last took a math class, so I have little to no memory of how to construct or go about formulating proofs. My tutor unfortunately is very and ...
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### How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
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Here is my question - Let $\|\cdot\|_s:\mathbb{R}^2\to\mathbb{R}^2$ be defined by: $$\|(x_1,x_2)\|_s=\left\{ \begin{array}{l l} \|(x_1,x_2)\|_2 & \quad \text{x_1x_2\geq 0}\\ \|(x_1,... 2answers 36 views ### p-norm on \mathbb{R}^n question How I can show that$$\lim_{p \to \infty} \|x\|_{p} = \max\{|x_1|, \; |x_2|, \; \cdot ,\; |x_n|\}$$if \mathbb{R}^n has the p-norm? p > 1 of course. Has anyone done this or know how to? I'm ... 1answer 309 views ### Norm of the Linear (Integral) Operator on C[0,1] There may be some similar questions being asked but I didn't see anything that was quite what I was looking for so I'll ask the question here. Let X = (C[0,1],||.||_{\max}) and T:X \to X = \int^{t}... 1answer 29 views ### Help proof regarding sequence in subset of Hilbert space I'm to prove the following: Let H be a Hilbert space, and let M be a non-empty convex subset of H. Suppose that (x_n) is a sequence in M such that  ||x_n|| \to d, where d= \inf \{||x||:... 1answer 154 views ### Show that \infty-norm and C^1-norm are not equivalent. Show that \infty-norm and C^1-norm are not equivalent. For the C^1([a,b],\mathbb{R}) space, show that \displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)| and \displaystyle ||g||_{C^1}=... 1answer 49 views ### Bounding Sobolev Norms *Below* Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ... 1answer 57 views ### Matrix one-norm and infinity-norm Help me please to find 3\times 3 matrices A and B under following conditions: \left \| A \right \|_{\infty }=4\left \| A \right \|_{1}, A \neq 0 \left \| B \right \|_{1}=4\left \| B \right \|... 1answer 85 views ### Show that \limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1 Let (x_n) be weakly convergent, but not norm convergent, sequence from a Banach space. Show that \limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1. Any help? 1answer 42 views ### Norm of the functional f(x)=\int_0^2 x(t).(t^2-1)dt in the space X=L_1(0,2). I am trying to calculate the norm of the functional f(x)=\int_0^2 x(t).(t^2-1)dt in the space X=L_1(0,2). What we have is: ||f(x)||\le ||x||_1.||t^2-1||_{\infty} by extended Hölder's inequality.... 1answer 53 views ### Question on the norms I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. \psi(x)=\sum_{i}a_{i}T_{i}(... 2answers 126 views ### Immediate consequence of the definition of Operator Norm. Explain ||Av|| \leq ||A||_{op}||v|| for every v \in V I was wondering why this is true. Wikipedia says it's an immediate consequence of the definition but I just do not follow. I am using the ... 2answers 27 views ### How to prove this inequality for operator and function How to prove this? \sum_{k=1}^{\infty}|(Tf)_k|^2\leq ||T||^2||f||^2 where T is an operator and a function f. (Tf)_k is the k-th coordinate of Tf. Should this involve Cauchy-Schwarz or the ... 1answer 157 views ### Prove triangle inequality of vector norm I am trying to show that ||x+y||_p \leq ||x||_p + ||y||_p where p is an integer larger than 1, but not infinity (I proved those cases already), and ||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}} ... 1answer 569 views ### What minimizes the Chebyshev Distance? For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ... 1answer 44 views ### An equation related to covariance matrix, square root of the matrix, and Euclidean norm. How can I prove this equation:$${ ({ x }^{ T }\Sigma x) }^{ 1/2 }={ \left\| { \Sigma }^{ 1/2 }x \right\| }_{ 2 }$$In which \Sigma  is a covariance matrix. I tried some numerical examples in ... 2answers 97 views ### How to calculate the norm? We have the vector :$$ w=(1,3,5,1,3,5,\ldots,1,3) \in \mathbb{R}^{3k-1},  and we want to calculate its norm $\|w\|$. Now I would like to know how the norm $\|w\|$ can be calculated.
Does inequality $\|A\|_2\leq \| |A|_m \|_2$ hold for all square matrices $A$ ? Where $|A|_m$ is also a square matrix, defined as $|A|_m := [|a_i,j|]$. Two examples are provided for the case that ...