I defined my norm as: \newcommand{\norm}[1]{\left\lVert#1\right\rVert}  And my bold norm as: \newcommand{\bnorm}[1]{\boldsymbol{\norm{#1}}}  But I want to write something like \bnorm{x^{(i)}} where I don't want the exponent to be bold. I tried \boldsymbol{\left\lVert} x \boldsymbol{\right\rVert}  but due to the closing brackets left doesn't find right and it gives an error. Any ideas?
121527
I want text surrounded by a bold-face \left\langle ... \right\rangle. But when I use \mathbf or \pmb, I get errors, and when I use \bm, it doesn't seem to do anything. Is there a command that does this?  I don't want to use \big, because I want this in a macro that I can use in many different places.
3916
I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):  In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $   I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.  Thanks!
48332
I'm looking for a reference for a matrix-norm inequality that I used in , which has a few equivalent forms.  I will use notation that applies to complex vector spaces with a sesquilinear inner product, but of course the same applies over real matrices.  The statement is as follows:     Take $A,B \in \Bbb F^{n \times n}$.  Then    $$\vert\operatorname{tr}(A^*B)\vert \leq \sigma_1(A)\sum_{i=1}^n \sigma_i(B) = \|A\| \operatorname{tr}|B|$$   where $\sigma_i$ denotes the $i$th singular value, $|B| = (B^*B)^{1/2}$, and $\|\cdot\|$ denotes the spectral norm (induced Euclidean norm).   I did manage to find some references, but they're overkill, and the texts themselves are not readily accessible to the faint of heart (Bhatia's text is dense and Pedersen's is not about matrices in particular).    A suitable reference would be greatly appreciated.
75296
Note that (1) all norms on finite dimensional vector space are equivalent.  And here (2) norms on vector space are equivalent iff they have same topology.   Question : How can we prove (2) ? Please recommend reference  Thank you in anticipation
62275
This question was part of my linear algebra assignment and I was unable to solv eit. SO, I am asking you for help.  Consider the vector space V of real polynomials of degree less than or equal to n. Fix distinct real numbers $a_0 , …, a_k$ . For $p \in V$ define $max${${|p(a_j)| : 0\leq j \leq k}$} . It defines a  norm on V if:  A only if k&lt;n B only if $k\geq n$ C if $k+1 \leq n $ D if $k\geq n+1$ I tried thinking on which result / reasoning I should use but I am clueless on when the norm will not be defined . Can you please help me through it?
4650
I was looking at an exercise where it asked the following:  $$\begin{array}{ccc} p&amp;q&amp;p\rightarrow q \\ T&amp;T&amp;T \\ &amp;\ldots \end{array}$$  So, for the third column, I just put $T$ which was correct but I didn't understand what $\rightarrow$ meant. I have seen $\implies$ but I haven't the arrow. Are they the same thing?  Thanks a bunch!
90442
I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius.  I have managed to solve find the constants for $||.||_{1}$ and  $||.||_{2}$ but I cannot see how to continue if I want to prove the following:  $||A||_{\infty} \leq \sqrt{n}||A||_{2}$   and the ones for Frobenius:  $||A||_{F} \leq \sqrt{n}||A||_{1}$  $||A||_{F} \leq \sqrt{n} ||A||_{2}$  Any help is appreciated. Thank you for your time
46973
A solution I saw somewhere suggested to use   \usepackage{bbold} $\mathbb{1}$   which works well and I get what I want. However, now other symbols used to represent real numbers, natural numbers, etc look ugly.    I'd like them to look (without using bbold)    I'd like to only use that package to achieve that strange double one. Is there any way I can do that?
16500
I am reading the book "characterizations of c*-algebras" by Doran and Belfi. On the page 299, the book reads "It can also be shown that if $xy=yx$, then $\nu(x+y)\le\nu(x)+\nu(y)$." The definition of $\nu$ is as follows: in normed algebra,  $$\nu(x)=\limsup_{n\to\infty}\|x^n\|^{\frac1n}$$  Note that the algebra is not assumed to be complete and is also not assumed to be complex scalar(we can have real scalars)  This is my trial: I just tried to mimic the proof of the Minkowski inequality, but failed.  Thank you
23801
Exercise: Prove that the operator norm of the set $S$ of all linear operators $L:R^n\to R^m$ defines a norm on $S$  Definition of norm: A positive function $\| .\|$ on a real vector space $V$ is a norm if   $\|v\|=0$ if and only if $v=0$ $\|\lambda v\|=|\lambda|\|v\|$ for all $\lambda\in R$ and $v\in V$ $\|v+w\|\le\|v\|+\|w\|$ for all $v,w\in V$   Definition of operator norm: $\|L\|=sup_{a\ne 0}\|La\|/\|a\|$ or $=sup_{\|a\|=1}\|La\|$  First part (successful, I think):   if $\|L\|=0$, we have $sup_{\|a\|=1}\|La\|=0$, so $L=0$. if $L=0$, we have $sup_{\|a\|=1}\|0a\|=0$   Second part (unsuccessful):   $\|\lambda L\|=sup_{\lambda a\ne 0}\|\lambda La\|/\|\lambda a\|=sup_{\lambda a\ne 0}\lambda \|La\|/\lambda \|a\|=sup_{a\ne 0}\|La\|/\|a\|\ne |\lambda|\|L\|$   Third part (successful, I think):   $\|L(v+w)\|=sup_{\|v+w\|=1}\|Lv+Lw\|\le sup_{\|v\|=1}\|Lv\|+sup_{\|w\|=1}\|Lw\|=\|v\|+\|w\|$   Suggestions? Thoughts? Thanks!
49916
We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + b\sqrt{2})(a - b\sqrt{2})|$ it's clear that $\nu (xy) = \nu(x) \nu(y)$? ... Can someone please explain to me how this is clear?
36992
Is it possible to automatically "add" \left and \right commands in front of all brackets? The source code should be left intact, but LaTeX should interpret every pair of brackets as if they were preceded by such commands. This is because adding \left and \right commands is time consuming and makes source code longer and less readable. I considered defining new commands such as \lb and \rb for \left( and \right), but it doesn't seem very elegant to me: what if I also want to add square brackets for example?  Any ideas? Thanks.
7738
I have this expression: |x-|x-|x-4||| Now, every | sign has the same size, but I'd like to make the outward pairs bigger.   Is it possible to do it automatically, or do I have to define the size of each | sign manually? It doesn't work with $\left| \right|$. In this case I have 3 abs signs. But what if there would be for example 12? Is it possible to do it automatically?  This works fine:   \[ \abs*{\VPhantomL x-\abs*{\VPhantomM x-\abs{x-4}}} \]     But the inline version doesn't:  $\abs*{\VPhantomL x-\abs*{\VPhantomM x-\abs{x-4}}}$
13762
I know that the  |  stands for the norm, however I see it sometimes with two lines, sometimes with one. Is there a difference? Do I need to take care when writing a formula?  So basically, is there a difference between     $|a|$ and $\|a\|$ ?   Is it the same when it comes to vectors instead of just numbers?
57449
We define p-norm in this way: $\|x\|_p = \{\sum ^N_j=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $    when $ p \to \infty $  How can I prove this ?
67825
I want to get the following:    I tried the following  \[    a(i,k) \leftarrow     \min\Big\{0, r(k,k) + \sum_{i^{'} | i^{'}\notin \{i,k\}}\max\{0, r(i{'},k)\}\Big\} \]   and got    How can I make the part under the summation extend to left and right without making a space between plus and max as in the first figure?
40189
In one of my papers I found a notation where 2 letters one under other are placed in brackets without any additional operation symbol - attached image (N m). What does this mean, is this a shortcut of summation?