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Installation Reference: https://gohugo.io/installation/ Git Go Configure the Site Use Git cmd Input commands: cd D:\HugoSites hugo new site MyNewSite git init Download your favorite theme on hugo website. Duplicate theme data into “theme” folder in your local repository. Duplicate “Content” and “hugo.yaml” in exampleSite into the main folder of your local repository. Delete “hugo.toml” and “content/post/rich-content”. Change the value of “theme” in “config.yaml” to the name of the theme folder. eg: ...

First and Foremost I wrote this post after studying point-set topology (we skipped the section devoted to metric spaces) and before studying functional analysis. And now, although only a very limited number of functional contact, it seems to be quite plain. Main Text Introduction The Arzelà–Ascoli theorem stands as a pivotal theorem in functional analysis. It provides a crucial link between the compactness of a function space and its properties of equicontinuity and uniform boundedness. This theorem, named after the Italian mathematicians Giuseppe Arzelà and Enrico Ascoli, has found applications in numerous areas of analysis, ranging from the existence of solutions to differential equations to complex analysis. ...

Tables Collection Topological Properties of common Metric Spaces Topological Properties of common Metric Spaces Metric Space Norm Distance Separability Completeness (Banach space for normed space) \(n\)-dimension Eucilidean space \((\mathbb{R}^n,d)\) $$\Vert x\Vert = \sqrt{\sum_{i=1}^{n} x_i^2}$$ $$ d(x,y) = \sqrt{\sum_{i=1}^{n} (x_i-y_i)^2} $$ ✓ ✓ Discrete metric space \((X,d_0)\) \(X\) is countable / $$d_0(x,y)=\begin{cases}0, x=y\\1, x\not=y\end{cases}$$ ✓ ✓ \(X\) is uncountable ✗ ✓ Space of continuous functions \(\mathscr{C}[a,b]\) $$\Vert f\Vert = \max\limits_{x\in [a,b]} \vert f(x)\vert$$ $$d(f,g) = \max\limits_{x\in [a,b]} \vert f(x)-g(x)\vert$$ ✓ ✓ $$\Vert f\Vert = \int_a^b \vert f(t)\vert \,\mathrm{d}x$$ $$d(f,g) = \int_a^b \vert f(t)-g(t)\vert \,\mathrm{d}x$$ ✓ ✗ Space of bounded sequences \(\ell^\infty\) $$\Vert x\Vert = \sup\limits_{i\geq 1} \vert x_i\vert$$ $$d(x,y) = \sup\limits_{i\geq 1} \vert x_i-y_i\vert$$ ✗ ✓ Variable exponent sequence space \(\ell^p\) $$\Vert x\Vert = \left(\sum_{i=1}^{\infty} \vert x_i\vert^p\right)^\frac{1}{p}$$ $$d(x,y) = \left(\sum_{i=1}^{\infty} \vert x_i-y_i\vert^p\right)^\frac{1}{p}$$ ✓ ✓ \(p\)-norm Lebesgue space \(L^p\) $$\Vert f\Vert = \left(\int_{[a,b]} \vert f(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ $$d(f,g) = \left(\int_{[a,b]} \vert f(x)-g(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ ✓ ✓

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