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Section 5.1:
The Quaternions is the set . Addition and multiplication are defined as follows:
o ⃗ ⃗ ⃗ ⃗
o ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗
We will make more definitions that are analogous to the complex numbers:
o ( ) ( )
o
Now some analogous facts follow:
Fact 1: Let , then ⃗
Fact 2:
is a division ring
(To prove associativity, note that (⃗ ) ⃗ ( ⃗ )
Fact 3: The function is a multiplicative norm on
(Hint: Verify that ̅)
Fact 4: There is a homeomorphism of metric spaces:
Some interesting facts:
It is possible to prove that the product is a sum of 2 squares by using thefact that . Similarly, one can show that the product
is a sum of 4 squares using the fact that the quaternion
norm is multiplicative.
Questions:
Is there an analogue for the equation in the division ring of quaternions?
More generally, is there “quaternion analysis” ?
Section 5.2:
Reminder 1: Let K be a division ring. An absolute value is a function such that:
o , - o , - o , -
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Reminder 2: Let V be a right vector space over a division ring K with an absolute value . Anorm is a function ‖ ‖
o ,‖‖ - o , ‖‖- o ,‖ ‖ ‖‖ ‖‖-
In all of the next facts, K is a division ring and belongs to {R,C,H}
Fact 1: If V is a finite dimensional normed right vector space over K, then all
norms on V are equivalent. (My understanding is that the metrics induced by the norms are
induce the same topology on V)
Proof idea:
Show that any norm (‖ ‖ is equivalent to the Euclidean norm(‖ ‖. Show that ,‖‖ ‖‖- by using the basis of . Show that ,‖‖ ‖‖- by
verifying that ‖ ‖ ‖ ‖ is continuous and by using the Heine Borel property of
Euclidean spaces.
Fact 2: Let V,W be normed right vector spaces such that V is finite dimensional over K. Let
be a right K-module homomorphism. Then, ‖ ‖ ‖ ‖ is
continuous.
Proof Idea: Let ‖ ‖ denote the Euclidean metric. Using Fact 1, deduce that ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ are homeomorphisms ( denote the identities in
the category Set). Use a basis of V to show that ‖ ‖ ‖ ‖ is continuous. Thus,
is continuous.
Fact 3: Let V be a normed n-dimensional vector space over K. There are homeomorphisms:
o ()
o ()
o ()
Proof Idea: Use Fact 1.
Fact 4:
Fact 5: There is a homeomorphism
Proof Idea:
Definition 1:
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Questions:
Does fact 1 hold for for infinite dimensional vector spaces over R? If the answer is yes, this
would imply that any two norms on an infinite dimensional vector space over R induce the samemetric topology.
Let V be a normed vector space over K (not necessarily finite dimensional) and W be a subspace
of V. What is a natural way of defining a norm on V/W (algebraic not topological notation) using
the norm of V. If possible, does the topology induced by the natural norm on V/W coincide with
the final topology with respect to the map that sends x to x+W ? (Due to fact 2, we
know that the final topology is finer than the topology induced by a norm defined on V/W).
Section 5.3:
Defintion 1: P(V)
Defintion 2: The fundamental map is (add a note about equivalence classes)
Fact 1: The fundamental map is open
Proof Idea: By applying the definition of the final topology, it suffices to show that” is
open” implies that “[,-] is open”. Use the properties of norm to show that:
[,-] ,
[,-]- Fact 2: Let U be a K-subspace of V. Then, P(U) is closed in P(V)
Proof Idea: Apply the definition of the final topology along with the fact that U-{0} is p-
saturated.
Fact 3: Let U,V,W be vector spaces over K such that V=U+W and . Then there is
a homeomorphism .
Proof Idea: Let ⃗ be a nonzero element of W. Verify that the function
with the action ⃗ ⃗ ⃗ is a homeomorphism. Fact 2 may be helpful.
Corollary: * +
Fact 4:
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