How to prove triangle inequality for euclidean norm on complex number?Cauchy–Schwarz inequality for complex numbersProving the reverse triangle inequality of the complex numbersz and w are two complex numbers prove the relationshipElementary-Looking Inequality on n Complex NumbersTriangle Inequality about complex numbers, special caseInequality in the complex spaceTriangle inequality for complex numbersGeneralize standard deviation to vectorsA third triangle inequality?How the prove that the equality in the triangle inequality holds only if $z = tw$ or $w = cz$ for some $t,c in mathbbR^nn$
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How to prove triangle inequality for euclidean norm on complex number?
Cauchy–Schwarz inequality for complex numbersProving the reverse triangle inequality of the complex numbersz and w are two complex numbers prove the relationshipElementary-Looking Inequality on n Complex NumbersTriangle Inequality about complex numbers, special caseInequality in the complex spaceTriangle inequality for complex numbersGeneralize standard deviation to vectorsA third triangle inequality?How the prove that the equality in the triangle inequality holds only if $z = tw$ or $w = cz$ for some $t,c in mathbbR^nn$
$begingroup$
We were asked to show that when:
$displaystyle Vert ZVert = left(sum_k=1^n (x_k+iy_k)(x_k-iy_k)right)^1/2$ that $Vert Z+WVert leq Vert ZVert+Vert WVert$ whenever $Z$ and $W$ are vectors in complex numbers holds.
Can someone help me to prove this please?
complex-numbers euclidean-geometry
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
$endgroup$
add a comment |
$begingroup$
We were asked to show that when:
$displaystyle Vert ZVert = left(sum_k=1^n (x_k+iy_k)(x_k-iy_k)right)^1/2$ that $Vert Z+WVert leq Vert ZVert+Vert WVert$ whenever $Z$ and $W$ are vectors in complex numbers holds.
Can someone help me to prove this please?
complex-numbers euclidean-geometry
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
$endgroup$
$begingroup$
Welcome to Math.SE! Please consider editing your question to include your thoughts on the problem, and/or showing what you have already tried.
$endgroup$
– Peter Woolfitt
Mar 21 '15 at 0:58
add a comment |
$begingroup$
We were asked to show that when:
$displaystyle Vert ZVert = left(sum_k=1^n (x_k+iy_k)(x_k-iy_k)right)^1/2$ that $Vert Z+WVert leq Vert ZVert+Vert WVert$ whenever $Z$ and $W$ are vectors in complex numbers holds.
Can someone help me to prove this please?
complex-numbers euclidean-geometry
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
$endgroup$
We were asked to show that when:
$displaystyle Vert ZVert = left(sum_k=1^n (x_k+iy_k)(x_k-iy_k)right)^1/2$ that $Vert Z+WVert leq Vert ZVert+Vert WVert$ whenever $Z$ and $W$ are vectors in complex numbers holds.
Can someone help me to prove this please?
complex-numbers euclidean-geometry
complex-numbers euclidean-geometry
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
edited Mar 21 '15 at 1:17
TravisJ
6,40831830
6,40831830
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
asked Mar 21 '15 at 0:55
Zhiyue JiangZhiyue Jiang
11
11
$begingroup$
Welcome to Math.SE! Please consider editing your question to include your thoughts on the problem, and/or showing what you have already tried.
$endgroup$
– Peter Woolfitt
Mar 21 '15 at 0:58
add a comment |
$begingroup$
Welcome to Math.SE! Please consider editing your question to include your thoughts on the problem, and/or showing what you have already tried.
$endgroup$
– Peter Woolfitt
Mar 21 '15 at 0:58
$begingroup$
Welcome to Math.SE! Please consider editing your question to include your thoughts on the problem, and/or showing what you have already tried.
$endgroup$
– Peter Woolfitt
Mar 21 '15 at 0:58
$begingroup$
Welcome to Math.SE! Please consider editing your question to include your thoughts on the problem, and/or showing what you have already tried.
$endgroup$
– Peter Woolfitt
Mar 21 '15 at 0:58
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint: the Cauchy-Schwarz inequality.
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
$endgroup$
add a comment |
$begingroup$
Assuming Cauchy-Schwarz's inequality $|langle z,wrangle| leq |z||w|$, you have: $$beginalign |z+w|^2 &= (z+w)overline(z+w) \ &= (z+w)(overlinez+overlinew) \ &= zoverlinez+zoverlinew+overlinezw+woverlinew \ &= |z|^2 + 2,rm Re(zoverlinew) + |w|^2 \ &leq |z|^2+2|langle z,wrangle|+|w|^2 \ &leq |z|^2 + 2|z||w|+|w|^2 \ &= (|z|+|w|)^2, endalign$$ and take roots.
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: the Cauchy-Schwarz inequality.
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
$endgroup$
add a comment |
$begingroup$
Hint: the Cauchy-Schwarz inequality.
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
$endgroup$
add a comment |
$begingroup$
Hint: the Cauchy-Schwarz inequality.
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
$endgroup$
Hint: the Cauchy-Schwarz inequality.
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
answered Mar 21 '15 at 0:58
Fan ZhengFan Zheng
1,312617
1,312617
add a comment |
add a comment |
$begingroup$
Assuming Cauchy-Schwarz's inequality $|langle z,wrangle| leq |z||w|$, you have: $$beginalign |z+w|^2 &= (z+w)overline(z+w) \ &= (z+w)(overlinez+overlinew) \ &= zoverlinez+zoverlinew+overlinezw+woverlinew \ &= |z|^2 + 2,rm Re(zoverlinew) + |w|^2 \ &leq |z|^2+2|langle z,wrangle|+|w|^2 \ &leq |z|^2 + 2|z||w|+|w|^2 \ &= (|z|+|w|)^2, endalign$$ and take roots.
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
$endgroup$
add a comment |
$begingroup$
Assuming Cauchy-Schwarz's inequality $|langle z,wrangle| leq |z||w|$, you have: $$beginalign |z+w|^2 &= (z+w)overline(z+w) \ &= (z+w)(overlinez+overlinew) \ &= zoverlinez+zoverlinew+overlinezw+woverlinew \ &= |z|^2 + 2,rm Re(zoverlinew) + |w|^2 \ &leq |z|^2+2|langle z,wrangle|+|w|^2 \ &leq |z|^2 + 2|z||w|+|w|^2 \ &= (|z|+|w|)^2, endalign$$ and take roots.
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
$endgroup$
add a comment |
$begingroup$
Assuming Cauchy-Schwarz's inequality $|langle z,wrangle| leq |z||w|$, you have: $$beginalign |z+w|^2 &= (z+w)overline(z+w) \ &= (z+w)(overlinez+overlinew) \ &= zoverlinez+zoverlinew+overlinezw+woverlinew \ &= |z|^2 + 2,rm Re(zoverlinew) + |w|^2 \ &leq |z|^2+2|langle z,wrangle|+|w|^2 \ &leq |z|^2 + 2|z||w|+|w|^2 \ &= (|z|+|w|)^2, endalign$$ and take roots.
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
$endgroup$
Assuming Cauchy-Schwarz's inequality $|langle z,wrangle| leq |z||w|$, you have: $$beginalign |z+w|^2 &= (z+w)overline(z+w) \ &= (z+w)(overlinez+overlinew) \ &= zoverlinez+zoverlinew+overlinezw+woverlinew \ &= |z|^2 + 2,rm Re(zoverlinew) + |w|^2 \ &leq |z|^2+2|langle z,wrangle|+|w|^2 \ &leq |z|^2 + 2|z||w|+|w|^2 \ &= (|z|+|w|)^2, endalign$$ and take roots.
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
answered Mar 21 '15 at 1:01
Ivo TerekIvo Terek
46.7k954146
46.7k954146
add a comment |
add a comment |
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– Peter Woolfitt
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