diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index aea4bb8..2959957 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -1,4 +1,4 @@ -site_name: My notes +site_name: Mathematics and physics wiki docs_dir: '../../docs/en' theme: @@ -110,6 +110,18 @@ nav: - 'Compactness': mathematics/functional-analysis/normed-spaces/compactness.md - 'Linear operators': mathematics/functional-analysis/normed-spaces/linear-operators.md - 'Linear functionals': mathematics/functional-analysis/normed-spaces/linear-functionals.md + - 'Inner product spaces': + - 'Inner product spaces': mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md + - 'Direct sums': mathematics/functional-analysis/inner-product-spaces/direct-sums.md + - 'Orthonormal sets': mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md + - 'Total sets': mathematics/functional-analysis/inner-product-spaces/total-sets.md + - 'Fourier series': + - 'Formalism': mathematics/functional-analysis/inner-product-spaces/fourier-series/formalism.md + - 'Convergence': mathematics/functional-analysis/inner-product-spaces/fourier-series/convergence.md + - 'Polynomials': + - 'Legendre polynomials': mathematics/functional-analysis/inner-product-spaces/polynomials/legendre-polynomials.md + - 'Hermite polynomials': mathematics/functional-analysis/inner-product-spaces/polynomials/hermite-polynomials.md + - 'Laguerre polynomials': mathematics/functional-analysis/inner-product-spaces/polynomials/laguerre-polynomials.md - 'Topology': - 'Fiber bundles': mathematics/topology/fiber-bundles.md - 'Calculus': diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/direct-sums.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/direct-sums.md new file mode 100644 index 0000000..3b1f3a9 --- /dev/null +++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/direct-sums.md @@ -0,0 +1,99 @@ +# Direct sums + +> *Definition 1*: in a metric space $(X,d)$, the **distance** $\delta$ from an element $x \in X$ to a nonempty subset $M \subset X$ is defined as +> +> $$ +> \delta = \inf_{\tilde y \in M} d(x,\tilde y). +> $$ + +In a normed space $(X, \|\cdot\|)$ this becomes + +$$ + \delta = \inf_{\tilde y \in M} \|x - \tilde y\|. +$$ + +> *Definition 2*: let $X$ be a vector space and let $x, y \in X$, the **line segment** $l$ between the vectors $x$ and $y$ is defined as +> +> $$ +> l = \{z \in X \;|\; \exists \alpha \in [0,1]: z = \alpha x + (1 - \alpha) y\}. +> $$ + +Using definition 2, we may define the following. + +> *Definition 3*: a subset $M \subset X$ of a vector space $X$ is **convex** if for all $x, y \in M$ the line segment between $x$ and $y$ is contained in $M$. + +This definition is true for projections of convex lenses which have been discussed in [optics](). + +We can now provide the main theorem in this section. + +> *Theorem 1*: let $X$ be an inner product space and let $M \subset X$ be a complete convex subset of $X$. Then for every $x \in X$ there exists a unique $y \in M$ such that +> +> $$ +> \delta = \inf_{\tilde y \in M} \|x - \tilde y\| = \|x - y\|, +> $$ +> +> if $M$ is a complete subspace $Y$ of $X$, then $x - y$ is orthogonal to $X$. + +??? note "*Proof*:" + + Will be added later. + +Now that the foundation is set, we may introduce direct sums. + +> *Definition 4*: a vector space $X$ is a **direct sum** $X = Y \oplus Z$ of two subspaces $Y \subset X$ and $Z \subset X$ of $X$ if each $x \in X$ has a unique representation +> +> $$ +> x = y + z, +> $$ +> +> for $y \in Y$ and $z \in Z$. + +Then $Z$ is called an *algebraic complement* of $Y$ in $X$ and vice versa, and $Y$, $Z$ is called a *complementary pair* of subspaces in $X$. + +In the case $Z = \{z \in X \;|\; z \perp Y\}$ we have that $Z$ is the *orthogonal complement* or *annihilator* of $Y$. Also denoted as $Y^\perp$. + +> *Proposition 1*: let $Y \subset X$ be any closed subspace of a Hilbert space $X$, then +> +> $$ +> X = Y \oplus Y^\perp, +> $$ +> +> with $Y^\perp = \{x\in X \;|\; x \perp Y\}$ the orthogonal complement of $Y$. + +??? note "*Proof*:" + + Will be added later. + +We have that $y \in Y$ for $x = y + z$ is called the *orthogonal projection* of $x$ on $Y$. Which defines an operator $P: X \to Y: x \mapsto Px \overset{\mathrm{def}}= y$. + +> *Lemma 1*: let $Y \subset X$ be a subset of a Hilbert space $X$ and let $P: X \to Y$ be the orthogonal projection operator, then we have +> +> 1. $P$ is a bounded linear operator, +> 2. $\|P\| = 1$, +> 3. $\mathscr{N}(P) = \{x \in X \;|\; Px = 0\}$. + +??? note "*Proof*:" + + Will be added later. + +> *Lemma 2*: if $Y$ is a closed subspace of a Hilbert space $X$, then $Y = Y^{\perp \perp}$. + +??? note "*Proof*:" + + Will be added later. + +Then it follows that $X = Y^\perp \oplus Y^{\perp \perp}$. + +??? note "*Proof*:" + + Will be added later. + +> *Lemma 3*: for every non-empty subset $M \subset X$ of a Hilbert space $X$ we have +> +> $$ +> \mathrm{span}(M) \text{ is dense in } X \iff M^\perp = \{0\}. +> $$ + +??? note "*Proof*:" + + Will be added later. \ No newline at end of file diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/convergence.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/convergence.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/formalism.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/formalism.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md new file mode 100644 index 0000000..c058f84 --- /dev/null +++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md @@ -0,0 +1,122 @@ +# Inner product spaces + +> *Definition 1*: a vector space $X$ over a field $F$ is an **inner product space** if an **inner product** $\langle \cdot, \cdot \rangle: X \times X \to F$ is defined on $X$ satisfying +> +> 1. $\forall x \in X: \langle x, x \rangle \geq 0$, +> 2. $\langle x, x \rangle = 0 \iff x = 0$, +> 3. $\forall x, y \in X: \langle x, y \rangle = \overline{\langle y, x \rangle}$, +> 4. $\forall x, y \in X, \alpha \in F: \langle \alpha x, y \rangle = \alpha \langle x, y \rangle$, +> 5. $\forall x, y, z \in X: \langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$. + +Similar to the case in normed spaces we have the following proposition. + +> *Proposition 1*: an inner product $\langle \cdot, \cdot \rangle$ on a vector space $X$ defines a norm $\|\cdot\|$ on $X$ given by +> +> $$ +> \|x\| = \sqrt{\langle x, x \rangle}, +> $$ +> +> for all $x \in X$ and is called the **norm induced by the inner product**. + +??? note "*Proof*:" + + Will be added later. + +Which makes an inner product space also a normed space as well as a metric space, referring to proposition 1 in normed spaces. + +> *Definition 2*: a **Hilbert space** $H$ is a complete inner product space with its metric induced by the inner product. + +Definition 2 makes a Hilbert space also a Banach space, using proposition 1. + +## Properties of inner product spaces + +> *Proposition 2*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space, then +> +> $$ +> \| x + y \|^2 + \| x - y \|^2 = 2\big(\|x\|^2 + \|y\|^2\big), +> $$ +> +> for all $x, y \in X$. + +??? note "*Proof*:" + + Will be added later. + +Proposition 2 is also called the parallelogram identity. + +> *Lemma 1*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space, then +> +> 1. $\forall x, y \in X: |\langle x, y \rangle| \leq \|x\| \cdot \|y\|$, +> 2. $\forall x, y \in X: \|x + y\| \leq \|x\| + \|y\|$. + +??? note "*Proof*:" + + Will be added later. + +Statement 1 in lemma 1 is known as the Schwarz inequality and statement 2 is known as the triangle inequality and will be used throughout the section of inner product spaces. + +> *Lemma 2*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space and let $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ be sequences in $X$, if we have $x_n \to x$ and $y_n \to y$ as $n \to \infty$, then +> +> $$ +> \lim_{n \to \infty} \langle x_n, y_n \rangle = \langle x, y \rangle. +> $$ + +??? note "*Proof*:" + + Will be added later. + +## Completion + +> *Definition 3*: an **isomorphism** $T$ of an inner product space $(X, \langle \cdot, \cdot \rangle)_X$ onto an inner product space $(\tilde X, \langle \cdot, \cdot \rangle)_{\tilde X}$ over the same field $F$ is a bijective linear operator $T: X \to \tilde X$ which preserves the inner product +> +> $$ +> \langle Tx, Ty \rangle_{\tilde X} = \langle x, y \rangle_X, +> $$ +> +> for all $x, y \in X$. + +As a first application of lemma 2, let us prove the following. + +> *Theorem 1*: for every inner product space $(X, \langle \cdot, \cdot \rangle)_X$ there exists a Hilbert space $(\tilde X, \langle \cdot, \cdot \rangle)_{\tilde X}$ that contains a subspace $W$ that satisfies the following conditions +> +> 1. $W$ is an inner product space isomorphic with $X$. +> 2. $W$ is dense in $X$. + +??? note "*Proof*:" + + Will be added later. + +Somewhat trivially, we have that a subspace $M$ of an inner product space $X$ is defined to be a vector subspace of $X$ taken with the inner product on $X$ restricted to $M \times M$. + +> *Proposition 3*: let $Y$ be a subspace of a Hilbert space $X$, then +> +> 1. $Y$ is complete $\iff$ $Y$ is closed in $X$, +> 2. if $Y$ is finite-dimensional, then $Y$ is complete, +> 3. $Y$ is separable if $X$ is separable. + +??? note "*Proof*:" + + Will be added later. + +## Orthogonality + +> *Definition 4*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space, a vector $x \in X$ is **orthogonal** to a vector $y \in X$ if +> +> $$ +> \langle x, y \rangle = 0, +> $$ +> +> and we write $x \perp y$. + +Furthermore, we can also say that $x$ and $y$ *are orthogonal*. + +> *Definition 5*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space and let $A, B \subset X$ be subspaces of $X$. Then $A$ is **orthogonal** to $B$ if for every $x \in A$ and $y \in B$ we have +> +> $$ +> \langle x, y \rangle = 0, +> $$ +> +> and we write $A \perp B$. + +Similarly, we may state that $A$ and $B$ *are orthogonal*. + diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md new file mode 100644 index 0000000..db11abc --- /dev/null +++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md @@ -0,0 +1,65 @@ +# Orthonormal sets + +> *Definition 1*: an **orthogonal set** $M$ in an inner product space $X$ is a subset $M \subset X$ whose elements are pairwise orthogonal. + +Pairwise orthogonality implies that $x, y \in M: x \neq y \implies \langle x, y \rangle = 0$. + +> *Definition 2*: an **orthonormal set** $M$ in an inner product space $X$ is an orthogonal set in $X$ whose elements have norm 1. + +That is for all $x, y \in M$: + +$$ + \langle x, y \rangle = \begin{cases}0 &\text{if } x \neq y, \\ 1 &\text{if } x = y.\end{cases} +$$ + +> *Lemma 1*: an orthonormal set is linearly independent. + +??? note "*Proof*:" + + Will be added later. + +In the case that an orthogonal or orthonormal set is countable it can be arranged in a sequence and call it can be called an *orthogonal* or *orthonormal sequence*. + +> *Theorem 1*: let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal sequence in an inner product space $(X, \langle \cdot, \cdot \rangle)$, then +> +> $$ +> \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 \leq \|x\|^2, +> $$ +> +> for all $x \in X$. + +??? note "*Proof*:" + + Will be added later. + +Theorem 1 is known as the Bessel inequality, and we have that $|\langle x, e_n \rangle|$ are called the Fourier coefficients of $x$ with respect to the orthonormal sequence $(e_n)_{n \in \mathbb{N}}$. + +## Orthonormalisation process + +Let $(x_n)_{n \in \mathbb{N}}$ be a linearly independent sequence in an inner product space $(X, \langle \cdot, \cdot \rangle)$, then we can use the **Gram-Schmidt process** to determine the corresponding orthonormal sequence $(e_n)_{n \in \mathbb{N}}$. + +Let $e_1 = \frac{1}{\|x_1\|} x_1$ be the first step and let $e_n = \frac{1}{\|v_n\|} v_n$ be the $n$th step with + +$$ + v_n = x_n - \sum_{k=1}^{n-1} \langle x_n, e_k \rangle e_k. +$$ + +## Properties + +> *Proposition 1*: let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal sequence in a Hilbert space $(X, \langle \cdot, \cdot \rangle)$ and let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence in the field of $X$, then +> +> 1. the series $\sum_{n=1}^\infty \alpha_n e_n$ is convergent in $X$ $\iff$ $\sum_{n=1}^\infty | \alpha_n|^2$ is convergent in $X$. +> 2. if the series $\sum_{n=1}^\infty \alpha_n e_n$ is convergent in $X$ and $s = \sum_{n=1}^\infty \alpha_n e_n$ then $a_n = \langle s, e_n \rangle$. +> 3. the series $\sum_{n=1}^\infty \alpha_n e_n = \sum_{n=1}^\infty \langle s, e_n \rangle e_n$ is convergent in $X$ for all $x \in X$. + +??? note "*Proof*:" + + Will be added later. + +Furthermore, we also have that. + +> *Proposition 2*: let $M$ be an orthonormal set in an inner product space $(X, \langle \cdot, \cdot \rangle)$, then any $x \in X$ can have at most countably many nonzero Fourier coefficients $\langle x, e_k \rangle$ for $e_k \in M$ over the uncountable index set $k \in I$ of $M$. + +??? note "*Proof*:" + + Will be added later. \ No newline at end of file diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/hermite-polynomials.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/hermite-polynomials.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/laguerre-polynomials.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/laguerre-polynomials.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/legendre-polynomials.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/legendre-polynomials.md new file mode 100644 index 0000000..e69de29 diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/total-sets.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/total-sets.md new file mode 100644 index 0000000..90e93d3 --- /dev/null +++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/total-sets.md @@ -0,0 +1,58 @@ +# Total sets + +> *Definition 1*: a **total set** in a normed space $(X, \langle \cdot, \cdot \rangle)$ is a subset $M \subset X$ whose span is dense in $X$. + +Accordingly, an orthonormal set in $X$ which is total in $X$ is called a total orthonormal set in $X$. + +> *Proposition 1*: let $M \subset X$ be a subset of an inner product space $(X, \langle \cdot, \cdot \rangle)$, then +> +> 1. if $M$ is total in $X$, then $M^\perp = \{0\}$. +> 2. if $X$ is complete and $M^\perp = \{0\}$ then $M$ is total in $X$. + +??? note "*Proof*:" + + Will be added later. + +## Total orthornormal sets + +> *Theorem 1*: an orthonormal sequence $(e_n)_{n \in \mathbb{N}}$ in a Hilbert space $(X, \langle \cdot, \cdot \rangle)$ is total in $X$ if and only if +> +> $$ +> \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \|x\|^2, +> $$ +> +> for all $x \in X$. + +??? note "*Proof*:" + + Will be added later. + +> *Lemma 1*: in every non-empty Hilbert space there exists a total orthonormal set. + +??? note "*Proof*:" + + Will be added later. + +> *Theorem 2*: all total orthonormal sets in a Hilbert space have the same cardinality. + +??? note "*Proof*:" + + Will be added later. + +This cardinality is called the Hilbert dimension or the orthogonal dimension of the Hilbert space. + +> *Theorem 3*: let $X$ be a Hilbert space, then +> +> 1. if $X$ is separable, every orthonormal set in $X$ is countable. +> 2. if $X$ contains a countable total orthonormal set, then $X$ is separable. + +??? note "*Proof*:" + + Will be added later. + +> *Theorem 4*: two Hilbert spaces $X$ and $\tilde X$ over the same field are isomorphic if and only if they have the same Hilbert dimension. + +??? note "*Proof*:" + + Will be added later. + diff --git a/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md b/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md index a011836..4a0307a 100644 --- a/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md +++ b/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md @@ -9,7 +9,7 @@ Also called a *normed vector space* or *normed linear space*. -> *Definition 2*: a norm on a vector space $X$ defines a metric $d$ on $X$ given by +> *Proposition 1*: a norm on a vector space $X$ defines a metric $d$ on $X$ given by > > $$ > d(x,y) = \|x - y\|, @@ -17,9 +17,13 @@ Also called a *normed vector space* or *normed linear space*. > > for all $x, y \in X$ and is called a **metric induced by the norm**. +??? note "*Proof*:" + + Will be added later. + Furthermore, there is a category of normed spaces with interesting properties which is given in the following definition. -> *Definition 3*: a **Banach space** is a complete normed space with its metric induced by the norm. +> *Definition 2*: a **Banach space** is a complete normed space with its metric induced by the norm. If we define the norm $\| \cdot \|$ of the Euclidean vector space $\mathbb{R}^n$ by @@ -58,7 +62,7 @@ This adaptation also works for $C$, $l^p$ and $l^\infty$, obviously. Obtaining t By definition, a subspace $M$ of a normed space $X$ is a subspace of $X$ with its norm induced by the norm on $X$. -> *Definition 4*: let $M$ be a subspace of a normed space $X$, if $M$ is closed then $M$ is a **closed subspace** of $X$. +> *Definition 3*: let $M$ be a subspace of a normed space $X$, if $M$ is closed then $M$ is a **closed subspace** of $X$. By definition, a subspace $M$ of a Banach space $X$ is a subspace of $X$ as a normed space. Hence, we do not require $M$ to be complete. @@ -72,7 +76,7 @@ Convergence in normed spaces follows from the definition of convergence in metri ## Convergent series -> *Definition 5*: let $(x_k)_{k \in \mathbb{N}}$ be a sequence in a normed space $(X, \|\cdot\|)$. We define the sequence of partial sums $(s_n)_{n \in \mathbb{N}}$ by +> *Definition 4*: let $(x_k)_{k \in \mathbb{N}}$ be a sequence in a normed space $(X, \|\cdot\|)$. We define the sequence of partial sums $(s_n)_{n \in \mathbb{N}}$ by > > $$ > s_n = \sum_{k=1}^n x_k, @@ -108,7 +112,7 @@ From the notion of absolute convergence the following theorem may be posed. ## Schauder basis -> *Definition 6*: let $(X, \|\cdot\|)$ be a normed space and let $(e_k)_{k \in \mathbb{N}}$ be a sequence of vectors in $X$, such that for every $x \in X$ there exists a unique sequence of scalars $(\alpha_k)_{k \in \mathbb{N}}$ such that +> *Definition 5*: let $(X, \|\cdot\|)$ be a normed space and let $(e_k)_{k \in \mathbb{N}}$ be a sequence of vectors in $X$, such that for every $x \in X$ there exists a unique sequence of scalars $(\alpha_k)_{k \in \mathbb{N}}$ such that > > $$ > \lim_{n \to \infty} \|x - \sum_{k=1}^n \alpha_k e_k\| = 0, @@ -165,7 +169,7 @@ As a first application of this lemma, let us prove the following. In particular, every finite dimensional normed space is complete. -> *Proposition 1*: every finite-dimensional subspace $M$ of a normed space $(X, \|\cdot\|)$ is a closed subspace of $X$. +> *Proposition 2*: every finite-dimensional subspace $M$ of a normed space $(X, \|\cdot\|)$ is a closed subspace of $X$. ??? note "*Proof*:" @@ -173,7 +177,7 @@ In particular, every finite dimensional normed space is complete. Another interesting property of finite-dimensional vector space $X$ is that all norms on $X$ lead to the same topology for $X$. That is, the open subsets of $X$ are the same, regardless of the particular choice of a norm on $X$. The details are as follows. -> *Definition 7*: a norm $\|\cdot\|_1$ on a vector space $X$ is **equivalent** to a norm $\|\cdot\|_2$ on $X$ if there exists $a,b>0$ such that +> *Definition 6*: a norm $\|\cdot\|_1$ on a vector space $X$ is **equivalent** to a norm $\|\cdot\|_2$ on $X$ if there exists $a,b>0$ such that > > $$ > \forall x \in X: a \|x\|_1 \leq \|x\|_2 \leq b \|x\|_1. @@ -181,7 +185,7 @@ Another interesting property of finite-dimensional vector space $X$ is that all This concept is motivated by the following proposition. -> *Proposition 2*: equivalent norms on $X$ define the same topology for $X$. +> *Proposition 3*: equivalent norms on $X$ define the same topology for $X$. ??? note "*Proof*:"