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\newtheorem*{requirements}{Requirements}

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\begin{document}

\mytitle[Invertibility and spectrum of the multiplication operator on $\ell^2$]%
{Intertibility and spectrum\\of the multiplication operator\\on the space of square-summable sequences}

\medskip

\begin{objectives}
Establish an invertibility criterion and calculate the spectrum
of the multiplication operator on the space $\ell^2$.
\end{objectives}

\begin{requirements}
Space $\ell^2$, canonical base of $\ell^2$,
multiplication operator on $\ell^2$ and its norm,
algebraic operations (sum, product by a scalar and product)
of two multiplication operators on $\ell^2$,
closure of a set in a metric space,
bounded linear operators,
norm of a bounded linear operator,
invertibility of diagonal matrices,
invertibility of a bounded linear operator,
spectrum of a bounded linear operator.
\end{requirements}

\bigskip\noindent
Denote by $\NaturalNumbers$ the set of the natural numbers starting with zero:
\[
\NaturalNumbers\eqdef\{0,1,2,\ldots\}.
\]

\bigskip

\begin{defn}[the space of the square-summable sequences of complex numbers]
\[
\ell^2\eqdef\ell^2(\NaturalNumbers)
\eqdef\biggl\{x\in\ComplexNumbers^\NaturalNumbers\colon\ 
\underbrace{\sum_{j\quad}\quad x_j\quad\hsp{3em}}_?\,\biggr\}.
\]
The space $\ell^2$ is a Hilbert space with respect to the inner product
\[
\langle x,y\rangle \eqdef \sum_{j\in\NaturalNumbers}\overline{x_j}\,y_j.
\]
This inner product is linear with respect to the second argument.
Many authors define the inner product to be linear with respect to the first argument.
\end{defn}

\begin{defn}[the space of the bounded sequences of complex numbers]
\[
\ell^\infty\eqdef\ell^\infty(\NaturalNumbers)
\eqdef\biggl\{x\in\ComplexNumbers^\NaturalNumbers\colon\ \bigubrace{9em}\,\biggr\}.
\]
\end{defn}


\clearpage
\section*{Canonical base of $\boldsymbol{\ell^2}$ (review)}

\medskip

\begin{defn}[canonical base of $\ell^2$]
For every $n\in\NaturalNumbers$, denote by $e_n$ the sequence
\[
e_n\eqdef\bigl(\delta_{n,j}\bigr)_{j\in\NaturalNumbers}.
\]
In the following exercises we shall see
that $(e_n)_{n\in\NaturalNumbers}$ is an orthonormal base of $\ell^2$.
\end{defn}

\bigskip

\begin{exercise}
Write the sequence $e_2$:\quad
$\displaystyle e_2=\bigl(\,\ubrace{2em}\,,\,\ubrace{2em}\,,\,\ubrace{2em}\,,\ubrace{2em}\,,\,\ubrace{2em}\,,\ \ldots\ \bigr)$.
\end{exercise}

\bigskip

\begin{exercise}
Write the following sequences:
\par\smallskip
\begin{align*}
-3e_0&=\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr),
\\[3ex]
7e_1&=\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr),
\\[3ex]
4e_0-5e_3&=\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr).
\end{align*}
\end{exercise}

\bigskip

\begin{exercise}
Write the following sequences in terms of the basic elements $e_n$:
\par\medskip
\[
\bigl(0,0,0,5,0,\ldots\bigr)=\ubrace{6em},
\qquad\qquad
\bigl(2,0,-4,0,0,\ldots\bigr)=\ubrace{6em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[orthonormality of the canonical base]
Let $m,n\in\NaturalNumbers$. Calculate the product:
\par\medskip
\[
\langle e_m,e_n\rangle = \hsp{20em}
\]
\end{exercise}

\clearpage

\begin{exercise}[norm of an element of the canonical base]
Let $n\in\NaturalNumbers$. Calculate the norm of $e_n$:
\par\medskip
\[
\|e_n\|_2=\ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[norm of a multiple of an element of the canonical base]
Let $n\in\NaturalNumbers$ and $\lambda\in\ComplexNumbers$.
Calculate the norm of $\lambda e_n$:
\par\medskip
\[
\|\lambda e_n\|_2=\ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x\in\ell^2$ and $n\in\NaturalNumbers$. Calculate:
\par\medskip
\[
\langle e_n,x\rangle = \hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Suppose that $x\in\ell^2$ and $\langle e_n,x\rangle=0$ for all $n\in\NaturalNumbers$.
Then
\par\medskip
\[
\ubrace{6em}
\]
According to general criteria of orthonormal basis in a Hilbert space,
it implies that the sequence $(e_n)_{n\in\NaturalNumbers}$ is an orthonormal base of $\ell^2$.
\end{exercise}

\bigskip

\begin{exercise}
Let $x\in\ell^2$.
Write the decomposition of $x$ in the base $(e_n)_{n\in\NaturalNumbers}$:
\par\medskip
\[
x=\sum_{n\in\NaturalNumbers}\ubrace{4em} e_n.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x\in\ell^2$ and $m\in\NaturalNumbers$ such that
\[
\forall n\in\NaturalNumbers\setminus\{m\}\qquad \langle e_n,x\rangle = 0.
\]
What can you say about $x$?
Write $x$ in terms of some basic elements $e_n$ and some inner products.
\end{exercise}

\vspace{6em}

\clearpage
\section*{Definition of the multiplication operator (review)}

\medskip
\begin{exercise}[component-wise product of sequences, review]
Let $x,y\in\ComplexNumbers^\NaturalNumbers$.
Write the definition of the \emph{component-wise product} of the sequences $x$ and $y$:
\[
\forall n\in\NaturalNumbers\qquad (xy)_n\eqdef \ubrace{5em}.
\]
In other words,
\[
xy\eqdef\bigl(\,\ubrace{5em}\,\bigr)_{n\in\NaturalNumbers}.
\]
\end{exercise}

\bigskip

\begin{exercise}[definition of the multiplication operator, review]
Let $a\in\ell^\infty$.
Define $M_a\colon\ell^2\to\ell^2$ by
\[
M_a x \eqdef \ubrace{5em},
\]
that is,
\[
\forall x\in\ell^2\qquad\forall j\in\NaturalNumbers\qquad
(M_a x)_j\eqdef \ubrace{5em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[action of the multiplication operator on the canonical base, review]
Let $a\in\ell^\infty$ and $n\in\NaturalNumbers$.
Calculate $M_a e_n$.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ and $n\in\NaturalNumbers$.
Calcule the $\ell^2$-norm of the sequence $M_a e_n$:
\par\medskip
\[
\|M_a e_n\|_2 =\ubrace{5em}.
\]
\end{exercise}

\vspace{6ex}


\clearpage
\section*{Supremum and infimum (review)}

\medskip

\begin{defn}
Denote by $\ExtendedRealNumbers$ the \emph{extended real line}
$\RealNumbers\cup\{-\infty,+\infty\}$ with the canonical order.
The additional elements $-\infty$ and $+\infty$ satisfy
\[
\forall\alpha\in\RealNumbers\qquad-\infty<a,\qquad\qquad\qquad
\forall\alpha\in\RealNumbers\qquad\alpha<+\infty,\qquad\qquad
-\infty<+\infty.
\]
\end{defn}

\medskip

\begin{exercise}[upper bound of a set of real numbers]
Let $A\subset\ExtendedRealNumbers$ and $\beta\in\ExtendedRealNumbers$.
Recall the definition:
\par\medskip
\[
\beta\quad\text{is an upper bound of}\quad A
\qquad\Longleftrightarrow\qquad
\ubrace{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $A\subset\ExtendedRealNumbers$ and $\gamma\in\ExtendedRealNumbers$. Then,
\par\medskip
\[
\gamma\quad\text{is not an upper bound of}\quad A
\qquad\Longleftrightarrow\qquad
\ubrace{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $A\subset\ExtendedRealNumbers$ and $\beta\in\ExtendedRealNumbers$.
What does mean the phrase ``$\beta$ is the supremum of $A$''?
Write the definition using the concept of the upper bounds.
(It is known that for every $A\subset\ExtendedRealNumbers$
there exists a unique supremum in $\ExtendedRealNumbers$.
We shall not prove this fact here.)
\par\medskip
\[
\beta=\sup(A)
\qquad\Longleftrightarrow\qquad
\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $A\subset\ExtendedRealNumbers$ and $\beta\in\ExtendedRealNumbers$.
What does mean the phrase ``$\beta$ is the supremum of $A$''?
Write the answer in terms of the quantifications $\forall$ and $\exists$
and some of the inequalities $<,>,\le,\ge$,
without mentioning explicitly the concept of upper bounds.
\par\medskip
\[
\beta=\sup(A)
\qquad\Longleftrightarrow\qquad
\begin{cases}
1)\quad\forall \alpha\in A\quad\ldots\hsp{10em}\vertstrut
\\[1ex]
2)\quad\forall \gamma<\beta\quad\ldots\hsp{5em}\vertstrut
\end{cases}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x=(x_n)_{n\in\NaturalNumbers}$ be a sequence in $\RealNumbers$ and $\beta\in\ExtendedRealNumbers$.
What does mean the phrase ``$\beta$ is the supremum of the sequence $x$''?
Write the answer in terms of the quantifications $\forall$ and $\exists$
and some of the inequalities $<,>,\le,\ge$.
\par\medskip
\[
\beta=\sup_{n\in\NaturalNumbers}x_n
\qquad\Longleftrightarrow\qquad
\begin{cases}
1)\quad\forall n\in\NaturalNumbers\quad\ldots\hsp{5em}\vertstrut
\\[1ex]
2)\ \hsp{15em}\vertstrut
\end{cases}
\]
\end{exercise}

\clearpage
\section*{Multiplication operator:\\
boundness and norm (review)}

\medskip

\begin{exercise}[definition of the supremum norm, review]
Let $a\in\ell^\infty$. Put
\[
\nu\eqdef\|a\|_\infty=\sup_{n\in\NaturalNumbers}|a_n|.
\]
Write the definition of $\nu$ using the quantifications $\forall$ and $\exists$
and some of the inequalities $<,>,\le,\ge$,
without mentioning explicitly the concept of upper bounds.
\end{exercise}

\vspace{14ex}

\begin{exercise}[idea of the upper bound for the norm of the multiplication operator, review]
Let $a\in\ell^\infty$ and $x\in\ell^2$.
Write an upper bound for the norm of $M_a x$.
\[
\|M_a x\|_2 \le \ubrace{6em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[idea of the lower bound for the norm of the multiplication operator, review]
Let $a\in\ell^\infty$.
Compare $\|M_a\|$ to $\|M_a e_n\|_2$.
\end{exercise}

\vspace{10ex}

\begin{exercise}[norm of the multiplication operator, review]
Let $a\in\ell^\infty$. Then
\[
\|M_a\|=\ubrace{3em}.
\]
\end{exercise}

\clearpage
\section*{Algebraic operations\\
with multiplication operators, review}

\medskip\noindent
In the following exercises
recall that the set of the multiplication operators
is closed under algebraic operations:
for example, the sum of two multiplication operators
results to be also a multiplication operator.

\medskip

\begin{exercise}[sum of two multiplication operators, review]
Let $a,b\in\ell^\infty$. Then
\par\medskip
\[
M_a + M_b = \ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[product of a multiplication operator by a scalar, review]
Let $a\in\ell^\infty$ and $\lambda\in\ComplexNumbers$. Then
\par\medskip
\[
\lambda M_a = \ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[product of two multiplication operators]
Let $a,b\in\ell^\infty$. Then
\par\medskip
\[
M_a M_b = \ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[adjoint to the multiplication operator]
Let $a\in\ell^\infty$. Then
\[
M_a^\ast = \ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[identity operator as a multiplication operator]
Denote by $I$ the identity operator on the space $\ell^2$.
Find a function $a$ such that $M_a = I$.
\end{exercise}

\vspace{8ex}


\clearpage
\section*{Range and closed range of a sequence}

\medskip

\begin{exercise}[definition of the closure of a set in a metric space, review]
Let $(X,d)$ be a metric space, $A\subset X$ and $b\in X$.
What does mean the phrase ``$b$ is a point of closure of $A$''?
Write the definition using quantifications and inequalities:
\par\smallskip
\[
b\in\closure(A)
\qquad\Longleftrightarrow\qquad
\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{defn}[range and closed range of a sequence of complex numbers]
Let $a\colon\NaturalNumbers\to\ComplexNumbers$.
Denote by $\Range{a}$ the range (the set of the values) of $a$:
\[
\Range{a}\eqdef\bigl\{z\in\ComplexNumbers\colon\quad \exists n\in\NaturalNumbers\quad z=a_n\bigr\}
\]
and by $\ClosedRange{a}$ the closure of the range of $a$:
\[
\ClosedRange{a}\eqdef\closure(\Range{a}).
\]
\end{defn}

\bigskip

\begin{exercise}
Let $a\colon\NaturalNumbers\to\ComplexNumbers$.
Write the definition of $\ClosedRange{a}$
using quantifications and inequalities:
\par\medskip
\[
w\in\ClosedRange{a}
\qquad\Longleftrightarrow\qquad
\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}[example]
Consider a sequence $a\colon\NaturalNumbers\to\ComplexNumbers$ is defined by
\par\medskip
\[
a_n=\bigl(1+(-1)^n\bigr)\frac{n}{n+1}.
\]
Calculate some first components of $a$.
\par\medskip
\[
a=\biggl(\,\hsp{2em}\,,\,\hsp{2em}\,,\,\hsp{2em}\,,\hsp{2em}\,,\,\hsp{2em}\,,\ \ldots\ \biggr).
\]
Find $\Range{a}$ and $\ClosedRange{a}$.
\end{exercise}

\vspace{15ex}


\clearpage
\section*{Injectivity and kernel of a linear operator (review)}

\medskip\noindent

\begin{exercise}[definition of injective function]
Let $X,Y$ be some sets and $f\colon X\to Y$ be a function.
What does mean the phrase ``$f$ is injective''?
\end{exercise}

\vspace{8ex}

\begin{exercise}[definition of the kernel of a linear operator]
Let $T\colon\ell^2\to\ell^2$ be a linear operator.
Then the \emph{kernel} (called also the \emph{null-space}) of $T$ is defined by:
\par\medskip
\[
\ker(T)\eqdef\bigl\{x\in\ell^2\colon\quad\ubrace{6em}\,\bigr\}.
\]
\end{exercise}

\bigskip

\begin{exercise}[criterium of the injectivity of linear operator]
Let $T\colon\ell^2\to\ell^2$ be a linear operator.
Prove that
\[
T\ \text{is injective}\qquad\Longleftrightarrow\qquad\ker(T)=\{\nullsequence\}.
\]
\end{exercise}

\vspace{20ex}

\begin{exercise}[the kernel of a bounded linear operator is closed]
Let $T\colon\ell^2\to\ell^2$ be a bounded linear operator.
Prove that its kernel $\ker(T)$ is a closed set (in the sense of the norm in $\ell^2$).
\end{exercise}



\clearpage
\section*{Invertibility of a bounded linear operator (review)}

\bigskip

\begin{exercise}
Let $T\colon\ell^2\to\ell^2$ be a bounded linear operator.
It is said that $T$ is \emph{invertible}
if there exists a bounded linear operator $S\colon\ell^2\to\ell^2$ such that
\par\medskip
\[
\ubrace{20em}
\]
It is known that if there exists such operator $S$, then it is unique.
In the case of existence,
$S$ is called the \emph{inverse operator} to $T$ and is denoted by $T^{-1}$.
\end{exercise}

\bigskip

\begin{exercise}
Let $T\colon\ell^2\to\ell^2$ be an invertible bounded linear operator.
Prove that $T$ is injective.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $T\colon\ell^2\to\ell^2$ be an invertible bounded linear operator.
Prove that for all $x\in\ell^2$,
\par\medskip
\[
\|Tx\|_2\ge\|T^{-1}\|^{-1}\|x\|_2.
\]
\end{exercise}

\vspace{14ex}

\begin{exercise}
Let $T\colon\ell^2\to\ell^2$ be a bounded linear operator.
Suppose that for all $\eps>0$ there exists a sequence $x\in\ell^2$
such that $\|x\|_2=1$ and $\|T x\|_2<\eps$.
Prove that $T$ is not invertible.
\end{exercise}


\clearpage
\section*{Invertibility of diagonal matrices (review)}

\medskip\noindent
The following exercises help to review some ideas
useful to study the invertibility of the multiplication operator.

\bigskip

\begin{exercise}[diagonal matrix as a multiplication operator, review]
Let $a\in\ComplexNumbers^3$.
Denote the components of $a$ by $a_0,a_1,a_2$.
Denote by $\diag(a)$ the diagonal matrix with diagonal entries $a_0,a_1,a_2$:
\[
\diag(a)\eqdef
\matr{ccc}{
a_0 & 0 & 0 \\
0 & a_1 & 0 \\
0 & 0 & a_2
}.
\]
Calculate the product of $\diag(a)$ by a general vector $x\in\ComplexNumbers^3$:
\[
\diag(a) x=
\matr{ccc}{
a_0 & 0 & 0 \\
0 & a_1 & 0 \\
0 & 0 & a_2
}
\matr{c}{ x_0 \\ x_1 \\ x_2 }
=\hsp{15em}
\]
Write the answer in a short form:
\quad
$\displaystyle\diag(a) x = \matrbig{\,\ubrace{6ex}\,}_{j=0}^2$.
\end{exercise}

\bigskip

\begin{exercise}[invertibility of a diagonal matrix, review]
Let $a\in\ComplexNumbers^3$.
Recall the criterim of the invertibility of the diagonal matrix $\diag(a)$:
\[
\diag(a)\ \text{is invertible}
\qquad\Longleftrightarrow\qquad
\ubrace{10em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[inverse to the diagonal matrix, review]
Let $a\in\ComplexNumbers^3$ such that $\diag(a)$ is invertible.
Write the formula for the inverse matrix:
\par\medskip
\[
\diag(a)^{-1}=
\matr{c|c|c}{
\vphantom{\displaystyle\int_0^1} \hsp{3em} & \hsp{3em} & \hsp{3em} \\\hline
\vphantom{\displaystyle\int_0^1} \hsp{3em} & \hsp{3em} & \hsp{3em} \\\hline
\vphantom{\displaystyle\int_0^1} \hsp{3em} & \hsp{3em} & \hsp{3em}
}.
\]
Using the notation $1/a=(1/a_0,1/a_1,1/a_2)$ write the result in a short form:
\par\medskip
\[
\diag(a)^{-1}=\ubrace{7em}.
\]
\end{exercise}



\clearpage
\section*{Invertibility of the multiplication operator}

\medskip

\begin{exercise}
Let $a\in\ell^\infty$.
Prove that the following conditions are equivalent:
\par\medskip
\[
0\notin\ClosedRange{a}
\qquad\Longleftrightarrow\qquad
\inf_{j\in\NaturalNumbers}|a_j|>0.
\]
\end{exercise}

\vspace{12ex}

\begin{exercise}
Let $a\in\ell^\infty$ such that $0\notin\ClosedRange{a}$.
Prove that $M_a$ is invertible.
(Construct the inverse operator.)
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ such that $0\in\Range{a}$.
Prove that $M_a$ is not injective and therefore not invertible.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ such that $0\in\ClosedRange{a}$.
Prove that for all $\eps>0$ there exists a sequence $x\in\ell^2$
such that $\|x\|_2=1$ and
\[
\|M_a x\|_2<\eps.
\]
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ such that $0\in\ClosedRange{a}$.
Prove that $M_a$ is not invertible.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$.
Write a criterion:
\par\medskip
\[
M_a\ \text{is invertible}
\qquad\Longleftrightarrow\qquad
\hsp{15em}
\]
\end{exercise}

\bigskip


\clearpage
\section*{Spectrum of the multiplication operator}

\medskip

\begin{exercise}
Let $A$ be a bounded linear operator in a complex Banach space $X$.
Recall the definition of the spectrum of $A$.
\par\medskip
\[
\spectrum(A)\eqdef\biggl\{\lambda\in\ComplexNumbers\colon\hsp{15em}\biggr\}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in\ell^\infty$.
Find the spectrum of $M_a$.
\end{exercise}

\vspace{12ex}

\begin{exercise}
Let $A$ be a bounded linear operator in a complex Banach space $X$ and $\lambda\in\ComplexNumbers$.
What does mean the condition ``$\lambda$ is an eigenvalue of $A$?
\end{exercise}

\vspace{5ex}

\begin{exercise}
Let $A$ be a bounded linear operator in a complex Banach space $X$ and $\lambda\in\ComplexNumbers$.
Recall the definition of the point spectrum of $A$:
\par\medskip
\[
\pointspectrum(A)\eqdef\biggl\{\lambda\in\ComplexNumbers\colon\hsp{15em}\biggr\}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in\ell^\infty$.
Find the point spectrum of the multiplication operator $M_a$.
\end{exercise}

\vspace{12ex}

\begin{exercise}[spectrum of the multiplication operator in terms of its point spectrum]
Let $a\in\ell^\infty$.
Express $\spectrum(M_a)$ in terms of $\pointspectrum(M_a)$.
\end{exercise}

\vspace{5ex}



\clearpage
\section*{Multiplication operator and canonical base}

\medskip

\begin{exercise}
Let $a\in\ell^\infty$, $x\in\ell^2$ and $n\in\NaturalNumbers$.
Calculate:
\par\medskip
\[
\langle e_n,M_a x\rangle = \hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in\ell^\infty$ and $x\in\ell^2$.
Prove that
\[
M_a x = \sum_{j\in\NaturalNumbers} a_j\langle e_j,x\rangle\,e_j.
\]
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ and $j\in\NaturalNumbers$.
Calculate:
\par\medskip
\[
\langle e_j,M_a e_j\rangle=\hsp{20em}
\]
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ and the sequence $(\lambda_j)_{j\in\NaturalNumbers}$ is defined by:
\[
\forall j\in\NaturalNumbers\qquad\lambda_j\eqdef\langle e_j,M_a e_j\rangle.
\]
Express $a$ through $\lambda_j$.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ and $j,k\in\NaturalNumbers$ such that $j\ne k$.
Calculate:
\par\medskip
\[
\langle e_j,M_a e_k\rangle=\hsp{20em}
\]
\end{exercise}

\vspace{10ex}


\label{endlabel}
\end{document}
