Topologies on product spaces of $\mathbb{R}$ and their relationships

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Topologies on product spaces of $\mathbb{R}$ and their relationships

In this post, I will summarise several topologies established on the product spaces of $\mathbb{R}$, i.e. $\mathbb{R}^n$, $\mathbb{R}^{\omega}$ and $\mathbb{R}^J$, as well as their relationships. Topologies on product spaces of $\mathbb{R}$…

James Munkres Topology: Theorem 20.3 and metric equivalence

Proof of Theorem 20.3 Theorem 20.3 The topologies on $\mathbb{R}^n$ induced by the euclidean metric $d$ and the square metric $\rho$ are the same as the product topology on $\mathbb{R}^n$. Proof: a) Prove the two metrics can mutually limit ea…

James Munkres Topology: Theorem 20.4

Theorem 20.4 The uniform topology on $\mathbb{R}^J$ is finer than the product topology and coarser than the box topology; these three topologies are all different if $J$ is infinite. Proof: a) Prove the uniform topology is finer than the product…

两个1/x类的广义函数

[转载请注明出处]http://www.cnblogs.com/mashiqi 2017/04/15 1.$\text{p.v.}\,\frac{1}{x}$ 因为$(x \ln x - x)' = \ln x$, 所以$\int_0^a \ln x \mathrm{\,d}x = \lim_{\epsilon \to 0^+} \int_\epsilon^a \ln x \mathrm{\,d}x = \lim_{\epsilon \to 0^+}(x \ln x - x)\big|_\eps…

parallelogram

The parallelogram law in inner product spaces Vectors involved in the parallelogram law. In a normed space, the statement of the parallelogram law is an equation relating norms: In an inner product space, the norm is determined using the inner produc…

If the parts of an organization (e.g., teams, departments, or subdivisions) do not closely reflect the essential parts of the product, or if the relationship between organizations do not reflect the r

https://en.wikipedia.org/wiki/Conway%27s_law…