Mixing
Bases in Hilbert Space 2019
$begingroup$
Prove that the set of functions
$C$ = { $f_{0}(x)= sqrt{frac{1}{pi}}$, $f_{n}(x)= sqrt{frac{2}{pi}}. cos(nx)$, $n$ $in$ $N$ }
is an orthonormal basis in $L^{2}(0, pi)$.
Hint: Use the facts that the set of continuous functions on compact interval are dense in $L^2$ and every
polynomial on compact interval is dense in the set of continuous functions (Stone-Weierstrass theorem).
You need to show that $overline{Sp(C)}$ =$L^2(0, pi)$.
I have a key but I do not understand this question, especially third hint. I feel sorry. If you have any clear solution for the question, please help me. Thanks.
functional-analysis quantum-mechanics
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
$endgroup$
add a comment |
$begingroup$
Prove that the set of functions
$C$ = { $f_{0}(x)= sqrt{frac{1}{pi}}$, $f_{n}(x)= sqrt{frac{2}{pi}}. cos(nx)$, $n$ $in$ $N$ }
is an orthonormal basis in $L^{2}(0, pi)$.
Hint: Use the facts that the set of continuous functions on compact interval are dense in $L^2$ and every
polynomial on compact interval is dense in the set of continuous functions (Stone-Weierstrass theorem).
You need to show that $overline{Sp(C)}$ =$L^2(0, pi)$.
I have a key but I do not understand this question, especially third hint. I feel sorry. If you have any clear solution for the question, please help me. Thanks.
functional-analysis quantum-mechanics
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
$endgroup$
$begingroup$
If you do not understand the question, I do not think that an answer will help you. You should first try to understand the question. In the last hint, $Sp(C)$ is the linear hull (span) of $C$ and the line denotes the closure.
$endgroup$
– gerw
Jan 17 at 7:43
add a comment |
$begingroup$
Prove that the set of functions
$C$ = { $f_{0}(x)= sqrt{frac{1}{pi}}$, $f_{n}(x)= sqrt{frac{2}{pi}}. cos(nx)$, $n$ $in$ $N$ }
is an orthonormal basis in $L^{2}(0, pi)$.
Hint: Use the facts that the set of continuous functions on compact interval are dense in $L^2$ and every
polynomial on compact interval is dense in the set of continuous functions (Stone-Weierstrass theorem).
You need to show that $overline{Sp(C)}$ =$L^2(0, pi)$.
I have a key but I do not understand this question, especially third hint. I feel sorry. If you have any clear solution for the question, please help me. Thanks.
functional-analysis quantum-mechanics
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
$endgroup$
Prove that the set of functions
$C$ = { $f_{0}(x)= sqrt{frac{1}{pi}}$, $f_{n}(x)= sqrt{frac{2}{pi}}. cos(nx)$, $n$ $in$ $N$ }
is an orthonormal basis in $L^{2}(0, pi)$.
Hint: Use the facts that the set of continuous functions on compact interval are dense in $L^2$ and every
polynomial on compact interval is dense in the set of continuous functions (Stone-Weierstrass theorem).
You need to show that $overline{Sp(C)}$ =$L^2(0, pi)$.
I have a key but I do not understand this question, especially third hint. I feel sorry. If you have any clear solution for the question, please help me. Thanks.
functional-analysis quantum-mechanics
functional-analysis quantum-mechanics
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
asked Jan 16 at 20:34
mathsstudentmathsstudent
386
386
$begingroup$
If you do not understand the question, I do not think that an answer will help you. You should first try to understand the question. In the last hint, $Sp(C)$ is the linear hull (span) of $C$ and the line denotes the closure.
$endgroup$
– gerw
Jan 17 at 7:43
add a comment |
$begingroup$
If you do not understand the question, I do not think that an answer will help you. You should first try to understand the question. In the last hint, $Sp(C)$ is the linear hull (span) of $C$ and the line denotes the closure.
$endgroup$
– gerw
Jan 17 at 7:43
$begingroup$
If you do not understand the question, I do not think that an answer will help you. You should first try to understand the question. In the last hint, $Sp(C)$ is the linear hull (span) of $C$ and the line denotes the closure.
$endgroup$
– gerw
Jan 17 at 7:43
$begingroup$
If you do not understand the question, I do not think that an answer will help you. You should first try to understand the question. In the last hint, $Sp(C)$ is the linear hull (span) of $C$ and the line denotes the closure.
$endgroup$
– gerw
Jan 17 at 7:43
add a comment |
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$begingroup$
If you do not understand the question, I do not think that an answer will help you. You should first try to understand the question. In the last hint, $Sp(C)$ is the linear hull (span) of $C$ and the line denotes the closure.
$endgroup$
– gerw
Jan 17 at 7:43