# Functional analysis/Vector spaces

This page includes the vector spaces used in the course.

## Defining: vector space

Be K{\textstyle \mathbb {K} } a field and V=(V, +){\textstyle V=(V, +)} a commutative group. It is called V{\textstyle V} a K{\textstyle \mathbb {K} }-vector space when an image is ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V} with (λ, v)↦λ⋅v{\textstyle (\lambda , v)\mapsto \lambda \cdot v}, is defined which meets the following axioms λ, μ∈K{\textstyle \lambda , \mu \in \mathbb {K} } and v, w∈V{\textstyle v, w\in V} arbitrary .
* (ES) 1⋅v=v{\textstyle 1\cdot v=v} (scalar multiplication with the neutral element of the field)
* (AMS) λ⋅(μ⋅v)=(λ⋅μ)⋅v.{\textstyle \lambda \cdot (\mu \cdot v)=(\lambda \cdot \mu )\cdot v.} (associative scalar multiplication)
* (DV) λ⋅(v+w)=λ⋅v+λ⋅w.{\textstyle \lambda \cdot (v+w)=\lambda \cdot v+\lambda \cdot w.} (distributive for vectors)
* (DS) (λ+μ)⋅v=λ⋅v+μ⋅v.{\textstyle (\lambda +\mu )\cdot v=\lambda \cdot v+\mu \cdot v.} (distributive for scalars)

## End-dimensional vector spaces 1

Be n∈N{\textstyle n\in \mathbb {N} }, then is
* Q$^n${\textstyle \mathbb {Q} ^{n}} a finite dimensional Q{\textstyle \mathbb {Q} }-vector space of dimension n{\textstyle n},
* R$^n${\textstyle \mathbb {R} ^{n}} a finite dimensional R{\textstyle \mathbb {R} }-vector space of dimension n{\textstyle n},
* C$^n${\textstyle \mathbb {C} ^{n}} a finite dimensional C{\textstyle \mathbb {C} }-vector space of dimension n{\textstyle n},

## Learning Tasks

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

## Finite dimensional vector spaces 2

Be m, n∈N{\textstyle m, n\in \mathbb {N} }, then is
* Mat(m×n, Q){\textstyle Mat(m\times n, \mathbb {Q} )} (m×n{\textstyle m\times n} matrices with components in Q{\textstyle \mathbb {Q} }) a finite dimensional Q{\textstyle \mathbb {Q} }-vector space of the dimension m⋅n{\textstyle m\cdot n},
* Mat(m×n, R){\textstyle Mat(m\times n, \mathbb {R} )} (m×n{\textstyle m\times n} matrices with components in R{\textstyle \mathbb {R} }) a finite dimensional R{\textstyle \mathbb {R} }-vector space of the dimension m⋅n{\textstyle m\cdot n},
* Mat(m×n, C){\textstyle Mat(m\times n, \mathbb {C} )} (m×n{\textstyle m\times n} matrices with components in C{\textstyle \mathbb {C} }) a finite dimensional C{\textstyle \mathbb {C} }-vector space of the dimension m⋅n{\textstyle m\cdot n},

## Infinite-dimensional vector spaces of functions 1

Be C([a, b], K){\textstyle {\mathcal {C}}([a, b], \mathbb {K} )} the set of constant (engl. continuous) functions of the interval [a.b]{\textstyle [a.b]} in the field K=Q, R, C{\textstyle \mathbb {K} =\mathbb {Q} , \mathbb {R} , \mathbb {C} } as a range of values. Then
* C([a, b], Q){\textstyle {\mathcal {C}}([a, b], \mathbb {Q} )} an infinite dimensional Q{\textstyle \mathbb {Q} }-vector space,
* C([a, b], R){\textstyle {\mathcal {C}}([a, b], \mathbb {R} )} an infinite dimensional R{\textstyle \mathbb {R} }-vector space,
* C([a, b], C){\textstyle {\mathcal {C}}([a, b], \mathbb {C} )} an infinite dimensional C{\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 (f, g)↦f+g:=h{\textstyle (f, g)\mapsto f+g:=h} and h(x):=f(x)+g(x){\textstyle h(x):=f(x)+g(x)} for all x∈[a, b]{\textstyle x\in [a, b]}.
The external feature is likewise defined by the multiplication of the function values with the scalar for each x∈[a, b]{\textstyle x\in [a, b]}rt:
    $\cdot :\mathbb {K} \times V\to V$ with (λ, f)↦λ⋅f:=h{\textstyle (\lambda , f)\mapsto \lambda \cdot f:=h} and h(x):=λ⋅f(x){\textstyle h(x):=\lambda \cdot f(x)} for all x∈[a, b]{\textstyle x\in [a, b]}.

## Vector Space of Continuous Functions

The compactness of the definition range [a, b]{\textstyle [a, b]} makes the space C([a, b], R){\textstyle {\mathcal {C}}([a, b], \mathbb {R} )} of the steady functions of [a, b]{\textstyle [a, b]} according to R{\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 (C(R, R), ‖⋅‖$_N${\textstyle ({\mathcal {C}}(\mathbb {R} , \mathbb {R} ), \|\cdot \|_{\mathbb {N} }} a local convex topological vector space.

## Infinite-dimensional vector spaces of sequences 3

Be K=Q, R, C){\textstyle \mathbb {K} =\mathbb {Q} , \mathbb {R} , \mathbb {C} )} a field, then designated
* K$^N$:={(a$_n$)$_{n∈N}$|a$_n$∈K für alle n∈N}{\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 K{\textstyle \mathbb {K} }.
* c$_{oo}$(K):={(a$_n$)$_{n∈N}$∈K$^N$|∃$_{n0∈N}$∀$_{n≥n0}$:a$_n$=0}{\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 K{\textstyle \mathbb {K} }, which are all components of sequence 0 from an index barrier.
* c$_o$(K):={(a$_n$)$_{n∈N}$∈K$^N$|lim$_{n→∞}$a$_n$=0}{\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
* c(K):={(a$_n$)$_{n∈N}$∈K$^N$|∃$_{ao∈K}$:lim$_{n→∞}$a$_n$=a$_o$}{\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 K{\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 K{\textstyle \mathbb {K} } be a normed vector space. We now consider consequences in the vector space K{\textstyle \mathbb {K} }3:
* K{\textstyle \mathbb {K} }4 is the set of the sequences in the vector space K{\textstyle \mathbb {K} }5, in which from an index cabinet all the sequence elements are equal to the zero vector from K{\textstyle \mathbb {K} }6.
* K{\textstyle \mathbb {K} }7 is the set of zero sequences, the sequences relating to the standard K{\textstyle \mathbb {K} }8 converging against the zero vector, i.e.:
    K{\textstyle \mathbb {K} }9
* ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}0 is the set of convergent consequences in ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}1, the consequences relating to standard ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}2 converging against vector ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}3, i.e.:
    ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}4
The follow-up spaces can be normalized (e.g. with ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}5)

## space of polynomial vector

Be ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}6 a body and ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}7 a normed ⋅:K×V→V{\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 ⋅:K×V→V{\textstyle \cdot :\mathbb {K} \times V\rightarrow V}9.
For a special (λ, v)↦λ⋅v{\textstyle (\lambda , v)\mapsto \lambda \cdot v}0, (λ, v)↦λ⋅v{\textstyle (\lambda , v)\mapsto \lambda \cdot v}1 is a linear combination of vectors of (λ, v)↦λ⋅v{\textstyle (\lambda , v)\mapsto \lambda \cdot v}2, wherein the coeffcients of the scalar multiplication potencies are (λ, v)↦λ⋅v{\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 $(V,d)$?