Functional analysis/Vector spaces

This page includes the vector spaces used in the course.

Defining: vector space

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)

End-dimensional vector spaces 1

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 ${\textstyle n}$,

Learning Tasks

• ''(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?

Finite dimensional vector spaces 2

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 ${\textstyle m\cdot n}$,

Infinite-dimensional vector spaces of functions 1

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.

Internal and external link to vector spaces of functions 2

The internal link is defined as follows:

$$+:V\times V\to V$$ 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:

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

Vector Space of Continuous Functions

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

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

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

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

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

Infinite-dimensional vector spaces of sequences 3

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 ${\textstyle \mathbb {K} }$.

Infinite-dimensional vector spaces of sequences 4

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

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

Infinite-dimensional vector spaces of sequence 5

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

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

space of polynomial vector

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

$$V[x]:=\left\{p\,\left|\,(p_{n})_{n\in \mathbb {N} }\in c_{oo}(V)\wedge p(x):=\sum _{n=0}^{\infty }p_{n}\cdot x^{n}\right.\right\}$$ 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.

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 $\mathbb {Q} ^{n}$, $\mathbb {R} ^{n}$ oder $\mathbb {C} ^{n}$. With $V=\mathbb {K} ^{\mathbb {N} }$ and $\mathbb {K} =\mathbb {Q} ,\mathbb {R} ,\mathbb {C} )$ 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$\lambda \in \mathbb {K}$:

$\cdot :\mathbb {K} \times V\to V$ 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 $\mathbb {R}$ as a Vector space over the field $\mathbb {Q}$. Is $(\mathbb {R} ,+,\cdot ,\mathbb {Q} )$ 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 $(\mathbb {R} ,+,\cdot ,\mathbb {Q} )$ has as intersection $U_{1}\cap U_{2}$ with $U_{2}:={\sqrt {5}}\mathbb {Q}$ and the intersection contains just $0\in \mathbb {R}$!
• 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:

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

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