Functional analysis/Vector spaces

This page includes the vector spaces used in the course.

Defining: vector space

Be $𝕂$ a field and $V=(V,+)$ a commutative group. It is called $V$ a $𝕂$-vector space when an image is $\cdot :𝕂\times V\to V$ with $(\lambda ,v)\mapsto \lambda \cdot v$, is defined which meets the following axioms $\lambda ,\mu \in 𝕂$ and $v,w\in V$ arbitrary .

• (ES) $1\cdot v=v$ (scalar multiplication with the neutral element of the field)
• (AMS) $\lambda \cdot (\mu \cdot v)=(\lambda \cdot \mu )\cdot v.$ (associative scalar multiplication)
• (DV) $\lambda \cdot (v+w)=\lambda \cdot v+\lambda \cdot w.$ (distributive for vectors)
• (DS) $(\lambda +\mu )\cdot v=\lambda \cdot v+\mu \cdot v.$ (distributive for scalars)

End-dimensional vector spaces 1

Be $n\in \mathbb{N}$, then is

• ${\mathbb{Q}}^n$ a finite dimensional $\mathbb{Q}$-vector space of dimension $n$,
• ${ℝ}^n$ a finite dimensional $ℝ$-vector space of dimension $n$,
• ${ℂ}^n$ a finite dimensional $ℂ$-vector space of dimension $n$,

Learning Tasks

• ''(Distinction between operations - $𝕂$-Algebra) What are the characteristics of a $𝕂$-vector space and a $𝕂$-Algebra? Distinguish between three types of multiplication in a $𝕂$-Algebra and identify in the defining properties of the $𝕂$ vector spaces or the $𝕂$ algebra according to these types of multiplication.
  • Multiplication in field $𝕂$,
  • scalar multiplication as a binary function from $𝕂\times V$ to $V$,
  • Multiplication of elements from vector space as an inner link in a $𝕂$ algebra,
• '(Multiplications - Hilbert space) Be $𝕂:=ℝ$ or $𝕂:=ℂ$. By which properties are different a $𝕂$-vector space and a Hilbert space over the field $𝕂$? Distinguish three operations in a $𝕂$-Hilbert space and compare the defining properties of a multiplication as an inner link in a $𝕂$-Algebra with the properties of a scalar product in an Hilbert space above the field $𝕂$. What similarities and differences do you notice?

Finite dimensional vector spaces 2

Be $m,n\in \mathbb{N}$, then is

• $Mat(m\times n,\mathbb{Q})$ ($m\times n$ matrices with components in $\mathbb{Q}$) a finite dimensional $\mathbb{Q}$-vector space of the dimension $m\cdot n$,
• $Mat(m\times n,ℝ)$ ($m\times n$ matrices with components in $ℝ$) a finite dimensional $ℝ$-vector space of the dimension $m\cdot n$,
• $Mat(m\times n,ℂ)$ ($m\times n$ matrices with components in $ℂ$) a finite dimensional $ℂ$-vector space of the dimension $m\cdot n$,

Infinite-dimensional vector spaces of functions 1

Be $𝒞([a,b],𝕂)$ the set of constant (engl. continuous) functions of the interval $[a.b]$ in the field $𝕂=\mathbb{Q},ℝ,ℂ$ as a range of values. Then

• $𝒞([a,b],\mathbb{Q})$ an infinite dimensional $\mathbb{Q}$-vector space,
• $𝒞([a,b],ℝ)$ an infinite dimensional $ℝ$-vector space,
• $𝒞([a,b],ℂ)$ an infinite dimensional $ℂ$-vector space.

Internal and external link to vector spaces of functions 2

The internal link is defined as follows:

$$+:V\times V\to V$$ with $(f,g)\mapsto f+g:=h$ and $h(x):=f(x)+g(x)$ for all $x\in [a,b]$.

The external feature is likewise defined by the multiplication of the function values with the scalar for each $x\in [a,b]$rt:

$$\cdot :𝕂\times V\to V$$ with $(\lambda ,f)\mapsto \lambda \cdot f:=h$ and $h(x):=\lambda \cdot f(x)$ for all $x\in [a,b]$.

Vector Space of Continuous Functions

The compactness of the definition range $[a,b]$ makes the space $𝒞([a,b],ℝ)$ of the steady functions of $[a,b]$ according to $ℝ$ with the standard

$$\Vert f\Vert :=\underset{a}{\overset{b}{\int }}|f(x)|dx$$

to a standardized vector space (see also norms, metrics, topology). With the semi-standards

$$\Vert f{\Vert }_n:=\underset{-n}{\overset{+n}{\int }}|f(x)|dx$$

becomes $(𝒞(ℝ,ℝ),\Vert \cdot {\Vert }_{\mathbb{N}}$ a local convex topological vector space.

Infinite-dimensional vector spaces of sequences 3

Be $𝕂=\mathbb{Q},ℝ,ℂ)$ a field, then designated

• ${𝕂}^{\mathbb{N}}:=\{(a_n{)}_{n\in \mathbb{N}}|a_n\in 𝕂\text{ für alle }n\in \mathbb{N}\}$ the following sequences are set in $𝕂$.
• $c_{oo}(𝕂):=\{(a_n{)}_{n\in \mathbb{N}}\in {𝕂}^{\mathbb{N}}|{\mathrm{\exists }}_{n_0\in \mathbb{N}}{\mathrm{\forall }}_{n\ge n_0}:a_n=0\}$, the sequences set in $𝕂$, which are all components of sequence 0 from an index barrier.
• $c_o(𝕂):=\{(a_n{)}_{n\in \mathbb{N}}\in {𝕂}^{\mathbb{N}}|\underset{n\to \mathrm{\infty }}{\lim }{a}_n=0\}$ set convergent sequences to 0
• $c(𝕂):=\{(a_n{)}_{n\in \mathbb{N}}\in {𝕂}^{\mathbb{N}}|{\mathrm{\exists }}_{a_o\in 𝕂}:\underset{n\to \mathrm{\infty }}{\lim }{a}_n=a_o\}$, the set of convergent consequences in $𝕂$.

Infinite-dimensional vector spaces of sequences 4

Let $𝕂=\mathbb{Q},ℝ,ℂ$ be a field, then we define the following vector spaces:

• ${\ell }_1(𝕂):=\{(a_n{)}_{n\in \mathbb{N}}\in {𝕂}^{\mathbb{N}}|\sum _{n=1}^{\mathrm{\infty }}|a_n|<\mathrm{\infty }\}$, the set of all sequences in $𝕂$, that are absolute convergent ${\ell }_1(𝕂)$ iss a normed vector space with the norm $\Vert a\Vert :=\sum _{n=1}^{\mathrm{\infty }}|a_n|$).
• ${\ell }_p(𝕂):=\{(a_n{)}_{n\in \mathbb{N}}\in {𝕂}^{\mathbb{N}}|\sum _{n=1}^{\mathrm{\infty }}|a_n{|}^p<\mathrm{\infty }\}$ is the space all sequences in $𝕂$, that absolute p -summable. For $1\le p<\mathrm{\infty }$ the space is a normed space. For $0<p<1$ the space is a metric space with the metric $d_p((a_n{)}_n,(b_n{)}_n):=\sum _{n=1}^{\mathrm{\infty }}|a_n-b_n{|}^p$, the topology can also be created with a $p$-norm $\Vert (a_n{)}_n{\Vert }_p:=\sum _{n=1}^{\mathrm{\infty }}|a_n{|}^p$
• ${\ell }_{\mathrm{\infty }}(𝕂):=\{(a_n{)}_{n\in \mathbb{N}}\in {𝕂}^{\mathbb{N}}|{\mathrm{\exists }}_{C>0}:\underset{n\in \mathbb{N}}{\sup }|a_n|<C\}$ is the set of all bounded sequences in $𝕂$ . ${\ell }_{\mathrm{\infty }}(𝕂)$ is a normed space.

Infinite-dimensional vector spaces of sequence 5

Let $𝕂=\mathbb{Q},ℝ,ℂ$ a field and $(p_n{)}_{n\in \mathbb{N}}$ a monotonic non-increasing sequence with $0<p_n\le 1$ for all $n\in \mathbb{N}$, the we denote

• $\ell (𝕂,(p_n{)}_{n\in \mathbb{N}}):=\{(a_k{)}_{k\in \mathbb{N}}\in {𝕂}^{\mathbb{N}}|\sum _{k=1}^{\mathrm{\infty }}|a_k{|}^{p_k}<\mathrm{\infty }\}$ as the set of all sequences in $𝕂$ for which the sequence ${(|a_k{|}^{p_k})}_{k\in \mathbb{N}}$ is absolute convergent.
• For the space $\ell (𝕂,(p_n{)}_{n\in \mathbb{N}})$ we define with the following $p$-seminorms $\Vert a{\Vert }_n=\sum _{k=1}^{\mathrm{\infty }}|a_k{|}^{p_n}$ for sequences $a=(a_k{)}_{k\in \mathbb{N}.}$
• $\ell (𝕂,(p_n{)}_{n\in \mathbb{N}})$ is a pseudoconvex vector space with the $p$-seminorm system $\Vert \cdot {\Vert }_{\mathbb{N}}$
• Please note, that for all $p$-seminorms the index $n$ for the the exponent $p_n$ is fixed for every index $k\in \mathbb{N}$ of the sequence.

Impact spaces in normed vector space

Let $𝕂$ be a normed vector space. We now consider consequences in the vector space $𝕂$3:

• $𝕂$4 is the set of the sequences in the vector space $𝕂$5, in which from an index cabinet all the sequence elements are equal to the zero vector from $𝕂$6.
• $𝕂$7 is the set of zero sequences, the sequences relating to the standard $𝕂$8 converging against the zero vector, i.e.:

$𝕂$9

• $\cdot :𝕂\times V\to V$0 is the set of convergent consequences in $\cdot :𝕂\times V\to V$1, the consequences relating to standard $\cdot :𝕂\times V\to V$2 converging against vector $\cdot :𝕂\times V\to V$3, i.e.:

$\cdot :𝕂\times V\to V$4

The follow-up spaces can be normalized (e.g. with $\cdot :𝕂\times V\to V$5)

space of polynomial vector

Be $\cdot :𝕂\times V\to V$6 a body and $\cdot :𝕂\times V\to V$7 a normed $\cdot :𝕂\times V\to V$8-vector space, then designated

$$V[x]:=\left\{p|(p_n{)}_{n\in \mathbb{N}}\in c_{oo}(V)\wedge p(x):=\sum _{n=0}^{\mathrm{\infty }}{p}_n\cdot x^n\right\}$$ sets of polynomials with coefficients in $\cdot :𝕂\times V\to V$9.

For a special $(\lambda ,v)\mapsto \lambda \cdot v$0, $(\lambda ,v)\mapsto \lambda \cdot v$1 is a linear combination of vectors of $(\lambda ,v)\mapsto \lambda \cdot v$2, wherein the coeffcients of the scalar multiplication potencies are $(\lambda ,v)\mapsto \lambda \cdot v$3 of a scaler 698-1047-172940832.

Binary operations and functions on vector spaces of sequences 4

The binary operations and functions on vector spaces of sequences are defined component-wise, analog to addition and scalar multiplication on the vector spacee ${\mathbb{Q}}^n$, ${ℝ}^n$ oder ${ℂ}^n$. With $V={𝕂}^{\mathbb{N}}$ and $𝕂=\mathbb{Q},ℝ,ℂ)$ the binary operation is defined with $a:=(a_n{)}_{n\in \mathbb{N}}$, $b:=(b_n{)}_{n\in \mathbb{N}}$ and $c:=(c_n{)}_{n\in \mathbb{N}}$ in the following way:

$+:V\times V\to V$ mit $(a,b)\mapsto a+b:=c$ und $c_n:=a_n+b_n$ für alle $n\in \mathbb{N}$.

The binary function of scalar multiplication is defined by the multiplication of the components of the sequence with the scalar$\lambda \in 𝕂$:

$\cdot :𝕂\times V\to V$ mit $(\lambda ,a)\mapsto \lambda \cdot a:=c$ and $c_n:=\lambda \cdot a_n$ for all $n\in \mathbb{N}$.

Learning Activities

• Consider the set of real numbers $ℝ$ as a Vector space over the field $\mathbb{Q}$. Is $(ℝ,+,\cdot ,\mathbb{Q})$ a finite dimensional or an infinite dimensional Vector space over the field $\mathbb{Q}$? Explain your answer!
• Prove, that the vector $v_1=3$ and $v_2=\sqrt{3}$ span a linear subspace $U_1$ in the $\mathbb{Q}$-vector space $(ℝ,+,\cdot ,\mathbb{Q})$ has as intersection $U_1\cap U_2$ with $U_2:=\sqrt{5}\mathbb{Q}$ and the intersection contains just $0\in ℝ$!
• Analyse the subset property of the following vector space of sequences and consider property of convergence of series, which are generated by the sequences with:

${\ell }^p(𝕂):=\{(x_n{)}_n\in {𝕂}^{\mathbb{N}}:\sum _{n=1}^{\mathrm{\infty }}|x_n{|}^p<\mathrm{\infty }\}$.

Identify the subset property between ${\ell }^1(𝕂)$ and $c_o(𝕂)$? Generalize this approach on ${\ell }^1(V)$ and $c_o(V)$ for normed spaces $(V,\Vert \cdot \Vert )$! Is this true for metric spaces $(V,d)$?

See also

• Banachalgebra
• Complex numbers
• nets (mathematics)
• Sequence space
• Cauchy Sequence
• Linear Figure
• Pythagoras theorem in (Pre-)Hilbert Spaces
• Ideal (Algebra)
• Topological Invertability Criteria
• Open Machine Learning

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