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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 $\... 3answers 3k views ### How to prove$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$such that for all ... 4answers 408 views ### A Euclidean space that is homeomorphic to a non-Euclidean space Is there a well-known example (preferably in dimensions 2 or higher) of two homeomorphic spaces: (1) a metric space with the Euclidean metric and (2) a metric space that is not Euclidean? 1answer 939 views ### Equivalent Norms on$\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$is ... 1answer 532 views ### Range of a degenerate integral operator is closed I am reading around about integral operators and I came across an interesting example that I could not figure out. This example comes from here. It is Example 5.27, I have typed it verbatim for your ... 3answers 336 views ### Prove that the set of$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 ... 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 ... 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} $However, I want ... 1answer 429 views ### Continuous map$\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\...
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 ...
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 ...