# Tagged Questions

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### Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1$)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1$) if they induce equivalent metrics. At first I thought this ...
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### How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
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### 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 ...
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### comparison of 3 topologies on C[0,1]

I have a ring of continuous functions from $[0,1]$ to $\Bbb R$. And two norms $C[0,1]\to\Bbb R$. One is supremum of $|f(x)|,$ the other the value of $\int_0^1|f(x)|$. Then I get a Cartesian product of ...
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### addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
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### Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it ...
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### Prove that this series converges?

I have a Banach space $X$ and a linear operator $A \in L(X)$. $A$ is bounded such that $||A|| <1$. I then have to show that $$log(I-A)=\sum_{n \ge 1} \frac {A^n}n$$ converges. All I can come up ...
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### Proving that Euclidean space having the infinity metric is a complete metric space (stuck)

I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space. I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $a,b$) ...