Infinite linear combination
$begingroup$
I don't know how to start the proof of the following statement:
Let $V$ a real vector space with inner product $langlecdot,cdotrangle$. Consider a subset $C={v_j}_Jsubset V$ such that $langle v_j,v_krangle = 0$ for $jneq k$, i.e. $C$ is pairwise orthogonal. Suppose that exists $u$ such that
$$u= sum_{i=1}^infty lambda_i v_i$$
Show that $uin overline{text{span}(C)}^{parallelcdotparallel}$ where $parallelspace xparallel = sqrt{langle x,xrangle}$ is the norm induced by its inner product.
Please, HELP! :(
real-analysis functional-analysis vector-spaces orthogonality
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
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add a comment |
$begingroup$
I don't know how to start the proof of the following statement:
Let $V$ a real vector space with inner product $langlecdot,cdotrangle$. Consider a subset $C={v_j}_Jsubset V$ such that $langle v_j,v_krangle = 0$ for $jneq k$, i.e. $C$ is pairwise orthogonal. Suppose that exists $u$ such that
$$u= sum_{i=1}^infty lambda_i v_i$$
Show that $uin overline{text{span}(C)}^{parallelcdotparallel}$ where $parallelspace xparallel = sqrt{langle x,xrangle}$ is the norm induced by its inner product.
Please, HELP! :(
real-analysis functional-analysis vector-spaces orthogonality
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
$endgroup$
add a comment |
$begingroup$
I don't know how to start the proof of the following statement:
Let $V$ a real vector space with inner product $langlecdot,cdotrangle$. Consider a subset $C={v_j}_Jsubset V$ such that $langle v_j,v_krangle = 0$ for $jneq k$, i.e. $C$ is pairwise orthogonal. Suppose that exists $u$ such that
$$u= sum_{i=1}^infty lambda_i v_i$$
Show that $uin overline{text{span}(C)}^{parallelcdotparallel}$ where $parallelspace xparallel = sqrt{langle x,xrangle}$ is the norm induced by its inner product.
Please, HELP! :(
real-analysis functional-analysis vector-spaces orthogonality
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
$endgroup$
I don't know how to start the proof of the following statement:
Let $V$ a real vector space with inner product $langlecdot,cdotrangle$. Consider a subset $C={v_j}_Jsubset V$ such that $langle v_j,v_krangle = 0$ for $jneq k$, i.e. $C$ is pairwise orthogonal. Suppose that exists $u$ such that
$$u= sum_{i=1}^infty lambda_i v_i$$
Show that $uin overline{text{span}(C)}^{parallelcdotparallel}$ where $parallelspace xparallel = sqrt{langle x,xrangle}$ is the norm induced by its inner product.
Please, HELP! :(
real-analysis functional-analysis vector-spaces orthogonality
real-analysis functional-analysis vector-spaces orthogonality
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
asked Dec 11 '18 at 1:09
Sergio IvanSergio Ivan
286
286
add a comment |
add a comment |
1 Answer
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Let $u_n=sum_{j=1}^n lambda_j v_j , $ then $u_n in text{span}(C)$ and $u_n to u$ as $ntoinfty.$ Therefore $uin overline{text{span}(C)}^{parallelcdotparallel}.$
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
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1 Answer
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$begingroup$
Let $u_n=sum_{j=1}^n lambda_j v_j , $ then $u_n in text{span}(C)$ and $u_n to u$ as $ntoinfty.$ Therefore $uin overline{text{span}(C)}^{parallelcdotparallel}.$
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
$endgroup$
add a comment |
$begingroup$
Let $u_n=sum_{j=1}^n lambda_j v_j , $ then $u_n in text{span}(C)$ and $u_n to u$ as $ntoinfty.$ Therefore $uin overline{text{span}(C)}^{parallelcdotparallel}.$
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
$endgroup$
add a comment |
$begingroup$
Let $u_n=sum_{j=1}^n lambda_j v_j , $ then $u_n in text{span}(C)$ and $u_n to u$ as $ntoinfty.$ Therefore $uin overline{text{span}(C)}^{parallelcdotparallel}.$
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
$endgroup$
Let $u_n=sum_{j=1}^n lambda_j v_j , $ then $u_n in text{span}(C)$ and $u_n to u$ as $ntoinfty.$ Therefore $uin overline{text{span}(C)}^{parallelcdotparallel}.$
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
answered Dec 11 '18 at 7:08
MotylaNogaTomkaMazuraMotylaNogaTomkaMazura
6,597917
6,597917
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