Linked Questions

3
votes
3answers
870 views

All norms on $\mathbb{R}$ are equivalent. [duplicate]

Let $X$ be a vector space. Two norms $\|\cdot\|,\|\cdot\|':X\to\mathbb{R}$ are equivalent, if there are constants $\alpha,\beta >0$, such that for all $x\in X$ holds: $\alpha\|x\|\leq \|x\|'\leq \...
4
votes
1answer
767 views

If any two norms on a vector space are equivalent then the space is finite-dimensional [duplicate]

I need to prove: If any two norms on a vector space are equivalent then the space is finite-dimensional. I am aware of the converse of this result that on a finite dimensional vector space any two ...
1
vote
3answers
69 views

Can any metric on $\Bbb R^n$ be bounded above and below for any other metric? [duplicate]

Let $d_1(x,y)$ and $d_2(x,y)$ be any two metrics on $\mathbb{R}^n$. Can it be shown that, $$c\cdot d_2(x,y) \le d_1(x,y) \le C\cdot d_2(x,y)$$ for all $x,y \in \mathbb{R}^n$ for some fixed positive ...
0
votes
1answer
24 views

How to prove that the following metrics are topologically equivalent [duplicate]

I have $d_p(x,y) = [\sum | x_i - y_i|^p]^{1/p}$ and $d_q(x,y) = [\sum | x_i - y_i|^q]^{1/q}$ metrics in $\mathbb{R}^n$ and I want to prove that they are equivalent. I already know that $d_{\infty}(x,y)...
31
votes
3answers
2k views

Proving a matrix inequality

Let $A, B \in \mathbb R^{m\times m}$ be symmetric positive semi-definite matrices. Is it true that $$\sup_{\|x\| = 1} \left| \|Ax\| - \|Bx\| \right| \geq c(m) \|A-B\|,$$ with $c(m) > 0$ and where $\...
12
votes
3answers
3k views

How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?

I want to prove the following theorem (no idea whether it has a name): Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
2
votes
4answers
408 views

A Euclidean space that is homeomorphic to a non-Euclidean space

Is there a well-known example (preferably in dimensions 2 or higher) of two homeomorphic spaces: (1) a metric space with the Euclidean metric and (2) a metric space that is not Euclidean?
4
votes
1answer
939 views

Equivalent Norms on $\mathbb{R}^d$ and a contraction

Suppose we have a norm $|| \cdot ||$ on $\mathbb{R}^d$ and a linear transformation $T$ which is a contraction in regards to $c \in [0,1)$. How can I prove that $\exists k\in \mathbb{N}$ s.t $T^k$ is ...
4
votes
1answer
532 views

Range of a degenerate integral operator is closed

I am reading around about integral operators and I came across an interesting example that I could not figure out. This example comes from here. It is Example 5.27, I have typed it verbatim for your ...
2
votes
3answers
336 views

Prove that the set of $n \times n$ matrices with determinant $1$ is unbounded closed with empty interior in $\mathbb{R}^{n^{2}}$.

Prove that the set of $n \times n$ matrices with determinant $1$ is unbounded closed with empty interior in $\mathbb{R}^{n^{2}}$. The aplication $\det$ is continuous, so the inverse image of a closed ...
0
votes
3answers
547 views

When are norms not equivalent?

There are a lot of questions here on showing that two norms are not equivalent. I understand that two norms may not be equivaelent from their proofs, however I do not understand why this happened in ...
3
votes
1answer
386 views

Inequivalent norms

On an infinite dimensional, do there exist two norms which are not equivalent? I actually know that on a infinite dimensional space, the number of inequivalent norms are $ 2^{dimX} $ However, I want ...
6
votes
1answer
429 views

Continuous map $\mathbb{R}^n\rightarrow\mathbb{R}^n$

When we say some map $\phi=(\phi_1,\ldots,\phi_n)$ is a continuous map $\mathbb{R}^n\rightarrow\mathbb{R}^n$ we really mean that each component $\phi_i$ is continuous as a function $\mathbb{R}^n\...
3
votes
1answer
255 views

Topology on the space of matrices

The vector space of $n\times n$ real matrices is isomorphic as a vector space to $\mathbb R^{n\times n}$. Does it follow that it is "the same" as the topological space $\mathbb R^{n\times n}$ with the ...
2
votes
1answer
569 views

Matrix function is continuous iff its components are continuous

Let $A: J\subseteq \mathbb{R} \to \mathbb{K}^{n \times n}$ be a function, and denote $A = (A_{ij}(x))$. Equip $J$ with the euclidean metric, and the space of $n\times n$ matrices with the ...

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