Functional Analysis

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. ...

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 \(L^p\) $$\Vert f\Vert = \left(\int_{[a,b]} \vert f(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ $$d(f,g) = \left(\int_{[a,b]} \vert f(x)-g(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ ✓ ✓

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 \(L^p\) $$\Vert f\Vert = \left(\int_{[a,b]} \vert f(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ $$d(f,g) = \left(\int_{[a,b]} \vert f(x)-g(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ ✓ ✓