# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

18 views

### Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \vert \vert T \vert \...
11 views

31 views

### Clarification of ideas concerning a quotient space.

Suppose I have a vector space $V$, and I identify $x\in V$ with $\lambda x\in V$, where $x\neq 0$ and $\lambda>0$, $\lambda\in\mathbb{R}$. I'm confused about two things: (1) Can I define a norm on ...
58 views

### Prove that an arbitrary norm is continuous. Is my proof correct?

Let $f: \mathbb{F}^n\rightarrow \mathbb{R}$ be defined by $f(a_1,\cdots, a_n)=\|\sum a_jv_j\|$. Show $f$ is continuous on $\mathbb{F}^n$. 1. $\|\cdot\|$ is an arbitrary norm on $\mathcal{V}$. ...
43 views

### Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
33 views

### Showing this a norm

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 ...
31 views

### Inequality between norm of function and it's derivative

There is a theorem: Let $f$ be a continuously differentiable, $2\pi$-periodic function. Given $\int_{-\pi}^{\pi} f(x) dx = 0$, I need to prove that $$||f|| \le \frac{\pi}{2} \cdot ||f'||.$$ Where ...
37 views

### differentiability of the norm of L^1

Let $N_1$ denote the natural norm of the functional space $L^1(\Omega)$, where $\Omega$ is an open domain of $R^n$: $$N(y)=\int_\Omega |y(x)| dx$$ I have the following question regarding $N_1$: ...
65 views

### Is there a generalisation of norm catering for $\|a\mathbf{v}\|=\|\mathbf{v}\|$?

I'm working with a function $p$ which gives a kind of "size" of the vectors in my vector space, and it has all the properties of a norm except that $$p(a\mathbf{v})=p(\mathbf{v}).$$ Ordinarily a norm ...
41 views

21 views

20 views

### Vectors in Normed Space Must Have Finite Length?

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces. But, I've never seen this listed explicitly as an ...
14 views

### Use operator norm to rigorously prove exp(ln(I + A)) = I + A

Show that $\exp(\ln(I + A)) = I + A$ when the operator norm of $A$ is less than 1. A similar question has been posted, Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?, but this does not offer a real ...
92 views

21 views

49 views

21 views

### Entrywise expression for L2 matrix norm

The matrix norm induced by the $\ell^2$ norm is known to be equal to the maximum singular value of the matrix. The matrix norms induced by the $\ell^1$ and $\ell^\infty$ norms admit simple ...