Functional analysis/Vector spaces

This page includes the vector spaces used in the course.

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

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

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

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

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

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

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

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

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

The internal link is defined as follows:

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

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

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

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

$${\displaystyle \|f\|:=\displaystyle \int _{a}^{b}|f(x)|\,dx}$$

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

$${\displaystyle \|f\|_{n}:=\displaystyle \int _{-n}^{+n}|f(x)|\,dx}$$

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

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

• ${\textstyle \mathbb {K} ^{\mathbb {N} }:=\{(a_{n})_{n\in \mathbb {N} }\,|\,a_{n}\in \mathbb {K} \,{\mbox{ für alle }}n\in \mathbb {N} \}}$ the following sequences are set in ${\textstyle \mathbb {K} }$.
• ${\textstyle c_{oo}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\exists _{n_{0}\in \mathbb {N} }\forall _{n\geq n_{0}}\,:\,a_{n}=0\}}$, the sequences set in ${\textstyle \mathbb {K} }$, which are all components of sequence 0 from an index barrier.
• ${\textstyle c_{o}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\lim _{n\to \infty }a_{n}=0\}}$ set convergent sequences to 0
• ${\textstyle c(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\exists _{a_{o}\in \mathbb {K} }\,:\,\lim _{n\to \infty }a_{n}=a_{o}\}}$, the set of convergent consequences in ${\textstyle \mathbb {K} }$.

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

• ${\displaystyle \ell _{1}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\sum _{n=1}^{\infty }|a_{n}|<\infty \}}$, the set of all sequences in ${\displaystyle \mathbb {K} }$, that are absolute convergent ${\displaystyle \ell _{1}(\mathbb {K} )}$ iss a normed vector space with the norm ${\displaystyle \|a\|:=\sum _{n=1}^{\infty }|a_{n}|}$).
• ${\displaystyle \ell _{p}(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\sum _{n=1}^{\infty }|a_{n}|^{p}<\infty \}}$ is the space all sequences in ${\displaystyle \mathbb {K} }$, that absolute p -summable. For ${\displaystyle 1\leq p<\infty }$ the space is a normed space. For ${\displaystyle 0<p<1}$ the space is a metric space with the metric ${\displaystyle d_{p}((a_{n})_{n},(b_{n})_{n}):=\sum _{n=1}^{\infty }|a_{n}-b_{n}|^{p}}$, the topology can also be created with a ${\displaystyle p}$-norm ${\displaystyle \|(a_{n})_{n}\|_{p}:=\sum _{n=1}^{\infty }|a_{n}|^{p}}$
• ${\displaystyle \ell _{\infty }(\mathbb {K} ):=\{(a_{n})_{n\in \mathbb {N} }\in \mathbb {K} ^{\mathbb {N} }\,|\,\exists _{C>0}\,:\,\sup _{n\in \mathbb {N} }|a_{n}|<C\}}$ is the set of all bounded sequences in ${\displaystyle \mathbb {K} }$ . ${\displaystyle \ell _{\infty }(\mathbb {K} )}$ is a normed space.

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

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

Let ${\textstyle \mathbb {K} }$ be a normed vector space. We now consider consequences in the vector space ${\textstyle \mathbb {K} }$3:

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

${\textstyle \mathbb {K} }$9

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

${\textstyle \cdot :\mathbb {K} \times V\rightarrow V}$4

The follow-up spaces can be normalized (e.g. with ${\textstyle \cdot :\mathbb {K} \times V\rightarrow V}$5)

Be ${\textstyle \cdot :\mathbb {K} \times V\rightarrow V}$6 a body and ${\textstyle \cdot :\mathbb {K} \times V\rightarrow V}$7 a normed ${\textstyle \cdot :\mathbb {K} \times V\rightarrow V}$8-vector space, then designated

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

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

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

${\displaystyle +:V\times V\to V}$ mit ${\displaystyle (a,b)\mapsto a+b:=c}$ und ${\displaystyle c_{n}:=a_{n}+b_{n}}$ für alle ${\displaystyle n\in \mathbb {N} }$.

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

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

• Consider the set of real numbers ${\displaystyle \mathbb {R} }$ as a Vector space over the field ${\displaystyle \mathbb {Q} }$. Is ${\displaystyle (\mathbb {R} ,+,\cdot ,\mathbb {Q} )}$ a finite dimensional or an infinite dimensional Vector space over the field ${\displaystyle \mathbb {Q} }$? Explain your answer!
• Prove, that the vector ${\displaystyle v_{1}=3}$ and ${\displaystyle v_{2}={\sqrt {3}}}$ span a linear subspace ${\displaystyle U_{1}}$ in the ${\displaystyle \mathbb {Q} }$-vector space ${\displaystyle (\mathbb {R} ,+,\cdot ,\mathbb {Q} )}$ has as intersection ${\displaystyle U_{1}\cap U_{2}}$ with ${\displaystyle U_{2}:={\sqrt {5}}\mathbb {Q} }$ and the intersection contains just ${\displaystyle 0\in \mathbb {R} }$!
• 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:

${\displaystyle \quad \ell ^{p}(\mathbb {K} ):=\left\{(x_{n})_{n}\in \mathbb {K} ^{\mathbb {N} }\,:\,\sum _{n=1}^{\infty }|x_{n}|^{p}<\infty \right\}}$.

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

• 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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• Date: 10/20/2024