Tagged Questions
3
votes
0answers
50 views
What are norms used for?
These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
2
votes
2answers
52 views
two topology questions (open set and equivalence)
Two metrics $d_1$ and $d_2$ are called equivalent if there exist positive constants $\alpha, \beta$ s.t $\forall x,y\in\mathbb R^n: \alpha d_2(x,y)\le d_1(x,y)\le\beta d_2(x,y)$
I already proved that ...
2
votes
3answers
65 views
Does a norm have to map to $\mathbb R$?
Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary.
...
2
votes
1answer
63 views
Cauchy-Schwarz for metrics with arbitrary signatures
When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
6
votes
0answers
71 views
Proof that $\|(a,b)\| \leq \|(c,d)\|$ if $0 \leq a \leq c$ and $0 \leq b \leq d$ [duplicate]
Possible Duplicate:
Is norm non-decreasing in each variable?
Let $\| \cdot \|$ be any norm on $\mathbb{R}^{2}$. Let $0 \leq a \leq c$ and $0 \leq b \leq d$. Show that $\|(a,b)\| \leq ...
5
votes
1answer
78 views
A metric on $\mathbb{R}^n$ such that $d(\lambda x, \lambda y)=|\lambda| d(x,y)$ which is not induced by a norm
Let $V=\mathbb{R}^n$.
Let $d:V \times V\rightarrow \mathbb{R}$ a metric on $\mathbb{R}^n$.
Assume that for any $x,y\in V$ and $\lambda \in \mathbb{R}$, we have $d(\lambda x, \lambda y) = ...
5
votes
1answer
259 views
triangle inequality for a certain norm
Let $d$ be a metric on a (say real) vector space $E$, with the property
$$d(x,x+cy)=|c|d(x,x+y)$$ for all $x,y\in E$ and scalars $c$. I am trying to prove that $x\mapsto d(x,0)$ defines a norm.
The ...
2
votes
1answer
134 views
A question on norm of error vector
Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
1
vote
1answer
81 views
What is the proper term for the entity that relates a vector space and a set?
One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
12
votes
2answers
2k views
Difference between metric and norm made concrete: The case of Euclid
This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me.
This time I am making ...
