Linked Questions

3 votes
3 answers
3k 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 \...
Cornman's user avatar
  • 11.1k
5 votes
1 answer
1k 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 ...
QED's user avatar
  • 12.6k
1 vote
3 answers
128 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 ...
John Jacob's user avatar
0 votes
1 answer
38 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)...
Silkking's user avatar
  • 971
31 votes
3 answers
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 $\...
Roberto Rastapopoulos's user avatar
12 votes
3 answers
4k 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 ...
Huy's user avatar
  • 6,674
15 votes
2 answers
6k views

Every linear operator $T:X \to Y$ on a finite-dimensional normed space is bounded

Exercise : Show that if $X$ is a finite-dimensional normed space and $Y$ is a normed space, then every linear operator $T:X \to Y$ is bounded. Attempt : Since $X$ is finite-dimensional, say $\dim(...
Rebellos's user avatar
  • 21.3k
3 votes
4 answers
565 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?
user156619's user avatar
5 votes
1 answer
2k 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 ...
Mariah's user avatar
  • 3,760
6 votes
2 answers
2k views

Lyapunov exponent for 2D map?

Identify the Lyapunov exponent for the cat map: $C(x,y) = (2x+y , x+y)$. I am very confused as to finding the Lyapunov exponent for a two-dimensional map. I've come across a resource that states $$\...
Brandon Barry's user avatar
4 votes
1 answer
1k 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 ...
user557's user avatar
  • 11.9k
2 votes
3 answers
1k 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 ...
Lucas's user avatar
  • 4,035
5 votes
2 answers
208 views

Estimate the coefficients of a polynomial against its maximum

For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (I.e., the above sum ranges over all ...
fsp-b's user avatar
  • 1,034
4 votes
1 answer
668 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 ...
nullUser's user avatar
  • 27.8k
2 votes
1 answer
2k 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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