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\begin{document}

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\section*{basis in functional analysis}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{concepts_of_basis_in_functional_analysis}{}\section*{{Concepts of basis in functional analysis}}\label{concepts_of_basis_in_functional_analysis}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{basis in functional analysis} is a [[linear basis]] that is compatible with the [[topological space|topology]] of the underlying [[topological vector space]]. Therefore this is sometimes also referred to as a ``topological basis'', but beware that this term is also used for referring to the unrelated concept of a ``[[basis for the topology]]''.

[[basis|Bases]] in [[linear algebra]] are extremely useful tools for analysing problems. Using a basis, one can often rephrase a complicated abstract problem in concrete terms, perhaps even suitable for a computer to work with. A basis provides a way of describing a [[vector space]] in a way that:

\begin{enumerate}%
\item Is complete: every point in the space can be described in this fashion.
\item Has no redundancies: the description of a point is unique.

\end{enumerate}
When translated into the language of linear algebra, we recover the key properties of a basis: that it be a spanning set and linearly independent.

In infinite dimensions, having a basis becomes more valuable as the spaces get more complicated. However, the notion of a basis also becomes complex because the question of what makes a description admits different answers depending on whether we want only finite sums, we allow sequences, or we want infinite sums.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{basis}\hypertarget{basis}{}
\begin{enumerate}%
\item We say that $B$ is a \textbf{[[Hamel basis]]} if:

\begin{enumerate}%
\item Every element of $v$ is a finite linear combination of elements of $B$,
\item If $v = \sum_{b \in B} \alpha_b b$ then the $\alpha_b$ are unique.

\end{enumerate}
Alternatively, $B$ is [[linearly independent subset|linearly independent]] and $\Span(B) = V$; in other words, the [[span]] of $B$ is $V$ but no [[proper subset]] of $B$ has this property.


\item We say that $B$ is a \textbf{topological basis} if:

\begin{enumerate}%
\item Every element $v \in V$ is a limit of a [[sequence]] or (more generally) a [[net]] of finite linear combinations of elements of $B$,
\item No element of $B$ is a limit of a sequence or net of finite linear combinations of the \emph{other} elements of $B$.

\end{enumerate}
Alternatively, $B$ is [[total subset|total]] (meaning that its span is [[dense subspace|dense]]) but no proper subset of $B$ is total.


\item We say that $B$ is a \textbf{Schauder basis} if:

\begin{enumerate}%
\item Every element of $v$ is a (possibly infinite) sum of scales of elements of $B$,
\item If $v = \sum_{b \in B} \alpha_b b$ then the $\alpha_b$ are unique.

\end{enumerate}


\end{enumerate}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{enumerate}%
\item In the presence of the [[axiom of choice]], Hamel bases always exist.


\item If $B$ is a topological basis, then $B$ has a dual basis. Since $B \setminus \{b\}$ is not total but $B$ is total, the closure of the span of $B \setminus \{b\}$ must be a codimension $1$ subspace, whence the kernel of a non-trivial continuous linear functional on $V$, say $f_b$. By scaling, this functional can be assumed to satisfy $f_b(b) = 1$. Since $B \setminus \{b\} \subseteq \ker f$, $f(b') = 0$ for all $b' \in B$, $b' \ne b$.


\item If $B$ is a Schauder basis then it is a topological basis and so, as mentioned, has a dual basis. Then the coefficients in the sum $v = \sum \alpha_b b$ must be given by evaluating the dual basis on $v$: $v = \sum f_b(v) b$.



\end{enumerate}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{enumerate}%
\item In $C([0,1],\mathbb{C})$ with the norm ${\|f\|} = \max\{{|f(t)|}\}$:

\begin{enumerate}%
\item The monomials are linearly independent and have dense span, but do not form a topological basis as there is a sequence of polynomials with no linear term converging to $t$.


\item The trigonometric polynomials do form a topological basis. The dual basis is given by taking the Fourier coefficients of a function. However, it is not a Schauder basis as there are continuous functions which are not the uniform limit of their Fourier series.


\item The following is a Schauder basis. Let $(d_n)$ be the sequence $\{0, 1, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \dots\}$. Define $f_n$ to be the piecewise-linear function with the property that: $f_n(d_n) = 1$ and $f_n(d_k) = 0$ for $k \lt n$, and $f_n$ has the least ``breaks''. Then $f_n$ forms a Schauder basis for $C([0,1],\mathbb{C})$. This is the classical \emph{Faber-Schader} basis.



\end{enumerate}


\end{enumerate}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Enflo, P. (1973). A counterexample to the approximation problem in Banach spaces. \emph{Acta Math.}, \emph{130}, 309--317.


\item Semadeni, Z. (1982). \emph{Schauder bases in Banach spaces of continuous functions} (Vol. 918). Lecture Notes in Mathematics. Berlin: Springer-Verlag.



\end{itemize}
category: functional analysis

[[!redirects basis in functional analysis]] [[!redirects bases in functional analysis]] [[!redirects basises in functional analysis]]

[[!redirects Schauder basis]] [[!redirects Schauder bases]] [[!redirects Schauder basises]]



\end{document}