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Definition (Nowhere dense set) A set $A$ in a topological space $X$ is nowhere dense if the complement of its closure is dense in $X$, i.e. $\overline 阅读全文
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This post summarizes the proof for Theorem 7.9 in Royden's "Real Analysis". Theorem 9 If $\langle X, \rho \rangle$ is an incomplete metric space, it i 阅读全文
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The swapping of integration and taking limit of the integrand, like \(\int \lim_{n \rightarrow \infty} f_n = \lim_{n \rightarrow \infty} \int f_n \), 阅读全文
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This is a mindmap made from my notes for the paper "Principles of boundary element methods", which is written by Martin Costabel. 阅读全文
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This article summarizes the famous Maxwell equations in different forms, namely, time-dependent formulation, time-harmonic formulation and their corre 阅读全文
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This article summarizes a list of existence and uniqueness theorems for variational problems from (Monk 2003), which are organized from simple to comp 阅读全文
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Let $\mathcal{X}$ and $\mathcal{Y}$ be Hilbert spaces. Let $A: \mathcal{X} \rightarrow \mathcal{Y}$ be a bounded and linear operator. Then $$A(\mathca 阅读全文
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许久以前，我读到了侯捷先生于《深入浅出MFC》一书中所写的“勿在浮砂筑高台”这句话，颇受警醒与启发。如今在工科领域已摸索多年，亦逐渐真切而深刻地认识到，若没有坚实、完整、细致的数学理论作为基石，任何技术工作与所谓的“唯象”理论研究也都无异于浮沙筑台，经不起举一反三的推敲与寻根究底的质疑。而这也像是不 阅读全文
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Exercise 22.6 Recall that \(\mathbb{R}_{K}\) denotes the real line in the \(K\)-topology. Let \(Y\) be the quotient space obtained from \(\mathbb{R}_K 阅读全文
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In James Munkres "Topology" Section 22, the quotient space is defined as below. Definition Let \(X\) be a topological space, and let \(X^*\) be a part 阅读全文