# 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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### What is the example of $L^p$ space which is not a Hilbert Space except $p=2$.

I know that $L^p$-norm satisfy the parallelogram law for $p=2$. But when $p$ is not equal to $2$ then it does not satify the parallelogram law and $L^p$ space is not Hilbert Space. For this I need a ...
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### Find $||\cdot||_2$ norm of block matrix [closed]

Compute the square norm $||\cdot||_2$ of matrix : \begin{bmatrix} O & I_n & \dots & O \\ I_n & O & \ddots & \\ \vdots & \ddots & \ddots & I_n \\ ...
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### Proving something is a norm

Let $a<b$ be real numbers and $X=C[a,b]$ be the space of continuous functions $f:[a,b] \rightarrow \mathbb R$ Prove that $||f||_1 = \sup _{t \in [a,b]} |f(t)|$ indeed defines a norm on $X$. The ...
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### Operator Norm: Why is this formulation so rarely seen?

There are many equivalent definitions for the norm $||T||$ of a linear operator $T:X\to Y$. Citing the Wikipedia and some books about functional analysis, the equivalence of definitions are often ...
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### Show that every norm is a $1$-lipschitz function

Let $\|\cdot\|_0$, a norm on $\mathbb{R}^n$. Show that the function $\|\cdot\|_0$ is $1$-lipschitz and hence, continuous. Meaning, I need to prove that: $$\big|\|x\|_0-\|y\|_0\big| \le \|x-y\|$$ ...
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### Definition of $L^p$ norm of a vector-valued function

If $u$ is a vector the definition of the discrete norm will be $$\|u\|_{l^p}=(\sum |u_i|^p)^{1/p},$$ If $u$ is a function, $$\|u\|_{L^p}=\left(\int|u|^p\right)^{1/p}$$ But when $u$ is a vector-valued ...
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### How did Von Neumann think of the formula for scalar product?

My question is on the idea behind Von Neumann's formula for the scalar product induced from a norm that satisfies the parallelogram law. $\langle x,y\rangle=\frac 14(\|x+y\|^2-\|x-y\|^2)$. Was it by ...
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### How can I prove that $\int_{a}^{b} |f(x)|dx$ defines norm on $C([a,b])$?

I need to show that $$\parallel f \parallel = \int_{a}^{b} |f(x)|dx$$ is a norm on $C([a,b])$. I need to show that $\|f\|$ meets the properties of a norm: positive distance, if all elements ...
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### Show that $\lVert A \rVert_T := \lVert T^{-1} A T \rVert$ is a subordinate (induced) norm

I saw the following claim in many places without proof: Given an induced norm $\lVert \cdot \rVert$, $$\lVert A \rVert_T := \lVert T^{-1} A T \rVert$$ is also an induced norm. All of the texts I ...
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### Matrix log norm

How is it that the matrix log norm: $$\lim_{\epsilon\to 0}={||I+\epsilon A||-1\over \epsilon}$$ is equal to $$\max\left( \lambda \left({A+A^T\over 2}\right)\right)$$ (the biggest eigenvalue)
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### Euclidean norm gives length even in $>3$ dimensions?

In $1,2,3$ dimensions I can simply make triangles and see that Euclidean norm gives me the distance between two points (i.e. the length of the vector from one point to the other). In higher dimensions,...
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### When is the Frobenius norm equal to the spectral radius?

I know that the spectral radius is $\rho(A) = max |\lambda_l| = \S_{max}^2$ and that the Frobenius norm is $||A|| = \sqrt{tr(A^*A)} = (\sum_{k}S_k^2)^{1/2}$, which means I want to find the matrix A ...
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### Does $\mathbf x\cdot \mathbf y = 0$ imply that $\lVert x+y\rVert_1 = \lVert x\rVert_1 + \lVert y\rVert_1$?

If x and y are orthogonal vectors and we define $\lVert x\rVert_1 =\sum^{n}_{j=1} |x_j|$, is it possible to express $\lVert x+y\rVert_1$ in terms of $\lVert x\rVert_1$ and $\lVert y\rVert_1$ ? So I ...
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### Why is $|x · y| ≤ ||x||_1||y||_∞$?

So let $||x||_∞ :=$ max $_{j=1,...,n}|x_j|$ and by Cauchy-Schwarz, $|x · y| ≤ ||x||_2||y||_2$ . Why then does $|x · y| ≤ ||x||_1||y||_∞$ ? I'm not sure how to show this.
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### Interpretation of the normalized squared Frobenius norm of correlation matrix

Let ${\bf R}$ be a $n\times n$ correlation matrix. I was wondering what is the interpretation of the following norm: $$\frac{1}{n}\|{\bf R}\|_F^2$$ I know that is equal ...
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### Show that every open subset can be represented as a union of at most countably many balls

Consider $\mathbb{R^n}$ with the $\infty$-norm. Show that every open subset can be represented as a union of at most countably many balls. This might have been answered before but I do not ...
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### How to prove Positivity (property of the Norm in Rn)

Assume v is a vector in Rn. I want to prove that: the magnitude of v >= 0, and magnitude of v = 0 if and only if vector v = 0 vector All I can think of is ...
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### There are infinitely many vectors such that $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$

I want to prove that there are infinitely many solutions in $3$-space for $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$ (where bold refers to vectors). My proof: What is wrong with ...
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### Are some of the axioms of a norm of a vector space unnecessary?

I have a homework problem where my task is to find out if some of the axioms of a norm of a vector space are unnecessary, meaning they can be derived from other axioms (I presume from the problem ...
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