Mixing
What is the complete definition of inner product space
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There are some contradicting and confusing definitions online and I am trying to figure out the general case definition.
For example it is seems that order between elements in the range of the inner product should be defined, but I never saw it being mentioned explicitly.
Must the range of the inner product be part of the field that is used to construct the vector space itself? For example it could be the vectors from the vector space, and since scalar vector multiplication is defined, there is no contradiction in the definition.
Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?
linear-algebra
asked Jan 15 at 13:22
ArtiumArtium
4881618
$endgroup$
add a comment |
$begingroup$
There are some contradicting and confusing definitions online and I am trying to figure out the general case definition.
For example it is seems that order between elements in the range of the inner product should be defined, but I never saw it being mentioned explicitly.
Must the range of the inner product be part of the field that is used to construct the vector space itself? For example it could be the vectors from the vector space, and since scalar vector multiplication is defined, there is no contradiction in the definition.
Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?
linear-algebra
asked Jan 15 at 13:22
ArtiumArtium
4881618
$endgroup$
$begingroup$
Typically, if not always, an inner product space is either a real or complex vector space (and the inner product always takes values in the scalar field).
$endgroup$
– David C. Ullrich
Jan 15 at 14:18
$begingroup$
"Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?" I think you mean the requirement is changed from bilinearity to sesquilinearity. The latter is general, because complex conjugation reduces on $Bbb R$ to the identity operation.
$endgroup$
– J.G.
Jan 15 at 15:19
add a comment |
$begingroup$
There are some contradicting and confusing definitions online and I am trying to figure out the general case definition.
For example it is seems that order between elements in the range of the inner product should be defined, but I never saw it being mentioned explicitly.
Must the range of the inner product be part of the field that is used to construct the vector space itself? For example it could be the vectors from the vector space, and since scalar vector multiplication is defined, there is no contradiction in the definition.
Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?
linear-algebra
asked Jan 15 at 13:22
ArtiumArtium
4881618
$endgroup$
There are some contradicting and confusing definitions online and I am trying to figure out the general case definition.
For example it is seems that order between elements in the range of the inner product should be defined, but I never saw it being mentioned explicitly.
Must the range of the inner product be part of the field that is used to construct the vector space itself? For example it could be the vectors from the vector space, and since scalar vector multiplication is defined, there is no contradiction in the definition.
Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?
linear-algebra
linear-algebra
asked Jan 15 at 13:22
ArtiumArtium
4881618
asked Jan 15 at 13:22
ArtiumArtium
4881618
asked Jan 15 at 13:22
ArtiumArtium
4881618
asked Jan 15 at 13:22
ArtiumArtium
4881618
asked Jan 15 at 13:22
ArtiumArtium
4881618
4881618
$begingroup$
Typically, if not always, an inner product space is either a real or complex vector space (and the inner product always takes values in the scalar field).
$endgroup$
– David C. Ullrich
Jan 15 at 14:18
$begingroup$
"Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?" I think you mean the requirement is changed from bilinearity to sesquilinearity. The latter is general, because complex conjugation reduces on $Bbb R$ to the identity operation.
$endgroup$
– J.G.
Jan 15 at 15:19
add a comment |
$begingroup$
Typically, if not always, an inner product space is either a real or complex vector space (and the inner product always takes values in the scalar field).
$endgroup$
– David C. Ullrich
Jan 15 at 14:18
$begingroup$
"Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?" I think you mean the requirement is changed from bilinearity to sesquilinearity. The latter is general, because complex conjugation reduces on $Bbb R$ to the identity operation.
$endgroup$
– J.G.
Jan 15 at 15:19
$begingroup$
Typically, if not always, an inner product space is either a real or complex vector space (and the inner product always takes values in the scalar field).
$endgroup$
– David C. Ullrich
Jan 15 at 14:18
$begingroup$
Typically, if not always, an inner product space is either a real or complex vector space (and the inner product always takes values in the scalar field).
$endgroup$
– David C. Ullrich
Jan 15 at 14:18
$begingroup$
"Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?" I think you mean the requirement is changed from bilinearity to sesquilinearity. The latter is general, because complex conjugation reduces on $Bbb R$ to the identity operation.
$endgroup$
– J.G.
Jan 15 at 15:19
$begingroup$
"Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?" I think you mean the requirement is changed from bilinearity to sesquilinearity. The latter is general, because complex conjugation reduces on $Bbb R$ to the identity operation.
$endgroup$
– J.G.
Jan 15 at 15:19
add a comment |
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$begingroup$
Typically, if not always, an inner product space is either a real or complex vector space (and the inner product always takes values in the scalar field).
$endgroup$
– David C. Ullrich
Jan 15 at 14:18
$begingroup$
"Is linearity part of the definition? When the field is complex numbers, it seems that the definition is altered a little bit. So what is the definition for the general case?" I think you mean the requirement is changed from bilinearity to sesquilinearity. The latter is general, because complex conjugation reduces on $Bbb R$ to the identity operation.
$endgroup$
– J.G.
Jan 15 at 15:19