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spaces in functional analysis, basic concepts related to topological space

terms from the architecture mathematicians seem to lie a lot. This time it's about the concept of space in functional analysis.


One of the most general concepts is that of topological space. It is also in algebraic geometry is applied, not merely "typical Analysis.


We first define the concept of topology . A topology T on a set M is a system of subsets of this set that satisfies certain properties. On the one hand, the amount
M be an element of T, but also the empty set. If two sets in T, so also is their intersection in T are. The union of countably many elements of T will be back in T. Coarse we have said that a set system, with respect to the average education and union as the operation is complete on the set. The attentive reader will see now that T is of average education or association a magma. That is, we have a basic algebraic structure that can start to something.

summarize:

is then (M, T) a topological space . All quantities are the elements of a topology, called open sets .

The complement

a set A is in a different set M defined as follows:

A subset A is called completed if its complement in M is an open set, so this is part of the topology.

Obviously that is, the empty set and M itself are closed sets, although they are open sets, because the complement of the empty set is M, an open set, and the complement of M is the empty set, which is open. In short:

Also, the association two closed sets A and B is again complete. Why? According to the de Morgan laws, the complement of the union of two sets is the average of their Komplementmengen. Both volumes are completed, ie their complements are open. The intersection of two open sets is open. So even the average of the two Komplementmengen is open. Thus the complement of the union of A and B is open.

Similarly, we realize, that the intersection of two closed sets is again closed. The corresponding de Morgan's rule is: The complement of the intersection of two sets is the union of complements of both sets.

The conclusion a subset N M is the average of all closed sets A in M that contain N as a subset.

The interior of a subset N is the union of all open sets in M contained in N as a subset.

A environment of a point m, which is an element in M is a set U, so that there is an open set O that contains m as an element and itself a subset of U. In short:

environments do not have to be open, but, consequently, always include an open environment.

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quadratic forms and orthogonal groups

A body is a division ring in which the Multiplication is commutative. A vector space
over a field K is a K - Links module.

Thus we see that the concept vector space in this sense, nothing new.
They say the fact that we are dealing with a module over a field.
When talking here of a K-vector space, then with K will mean the body.
Let W be a submodule of a K-vector space V. Then W is a subspace

of V. Let U, W subspaces vonV. Then we define their sum U + W as follows:

It is even said that U + W is a subspace of V.
If U + W = V and
we write
and call V the direct sum of U and W.

So let V be a K-vector space. We now consider representations of V to K, which have particularly important structural properties, namely linearity.

A linear form is a mapping from V to K, which is linear, ie for which:
The set of all linear forms on a given vector space V forms its dual space .

A bilinear is a picture
the linear in each variable is.

A quadratic form on V is a mapping
with the property:
and to apply that
is a bilinear form.
We call this the bilinear form b polar form of Q and say that Q is polarized to b.

A quadratic form is nondegenerate if its polar form b has the property
.

Let V be a vector space over a field K and let B be a bilinear form on V. We call the set
the radical of b.
b is non-degenerate if and if
A vector space V, equipped with a nondegenerate quadratic form Q is called a orthogonal geometry .

A subspace W is called non-degenerate if
A singular vector is a
A non-zero vector u is isotropic if
A subspace is called W completely isotropic if
This means W is a subset of the radical of b is in W.

A pair of isotropic vectors (u, v)
a hyperbolic Pair, and the amount
is hyperbolic line.
Recall P (V) is the projective geometry of V.

Is
and also
we write and say
: V is orthogonal direct sum of U and W.

Why all this? Now there is a nice statement that we want to formulate a lemma. A
Lemma is a phrase that is required as a prerequisite to formulate a mathematical theorem. The proposition to prove the the following lemma helps is the set of Witt, which we consider in this context does not, however, be. We need the lemma as a separate sentence.

Lemma: Let V
provided by Q, an orthogonal geometry. Let W be a non-degenerate two-dimensional subspace of V that contains a bilinear form with respect to the isotropic vector u b. Then W is the product of a hyperbolic pair (u, v), which can be chosen v such that Q (v) = 0 Therefore, there is then
In the proof we give here, because we do not need another lemma.
He can read about in the very recommendable book by Donald E. Taylor, "The geometry of the Classical Groups".

If V is an orthogonal geometry over a field K with q elements and contains a singular vector, we can write to the above lemma:
are being
hyperbolic lines and W contains no singular vector.
The number m is Witt index of V.

sentence Each vector space of dimension at least 3 over a finite field has a singular vector.
(here without proof, proof again at Taylor "The geometry of the Classical Groups" p.138)

the dimension of W under the above conditions are only 0.1 or 2 after that game. Witt's theorem then says that V up to isomorphism (bijective linear maps) by m W and is unique.

We want to write down the hyperbolic lines of generators. That is
Let V and W two K-vector spaces. A picture
is K-linear map, or homomorphism if Be
V is a K-vector space and f is a homomorphism of V into itself
f is a bijective homomorphism, then f is called a automorphism of V.
Since f is bijective, there is an inverse linear transformation of f, ie a linear map g so that the sequence of execution is f and g is the identity.
It is said therefore that f is invertible, or that is f regular.
The set of all regular linear maps of V, with the concatenation of images as a link, called the general linear group of V and we write for GL (V).

That the GL (V) is a group, the reader may verify himself.

Now we can formulate what is the orthogonal group.
Let V be a K-vector space and Q a nondegenerate quadratic form. The
orthogonal group, which is associated with V and Q is the amount

provided with the successive application of images as a link.

Let G be a group and a, b be elements in G. Then we call
the commutator of a and b.
We call the subgroup generated by all commutators
the commutator subgroup of G. The commutator

of O (V, Q) is denoted by
Getting back to the orthogonal geometry V over a finite field K, which consists of q elements. There is, as described above
where:
If W = 0, then the dimension of V is the double of Witt-index which is 2m. To express this situation, we write for the orthogonal group
and their commutator
The quadratic form Q in this case is of the form:

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by the Group to the building

We now know from her previous post, what a group is.
Here there is a definition for the group, which sounds a bit different, but
is equivalent to the others.

just to avoid misunderstandings: This Muliplikation can also be an addition, and one can therefore also be 0.

An important property that can occur when magma is the Commutativity .
A magma M is called commutative, if true:
If M is commutative and a group, we call M abelian or an abelian group.

So far we only considered a set with a link. But we can assign a set and two links.

Let M be a set with two operations

also be covered by the two distributive, namely

Then
a half ring .

A ring is a semiring
in the (R, +) an abelian group. The neutral element of (R, +) is denoted by "0". In most cases, is also required that the ring is a unit element "1" contains so
a neutral element of multiplication. The ring is then unitary .
are we when we speak of struggle, always talking about unitary rings.

A ring is called a division ring (or division ring) if R is a ring and if he has any nonzero element is a multiplicative inverse.
This is equivalent to saying that the ring is without the zero, a group with respect to the Multiply.

that was not enough. Over algebraic structures can be re-defined algebraic structures.

Let R be a ring. Let M be an additive abelian group. There is an additional link

also may apply the following:

Then M is an R-module links.

A right R-module M is defined the same, but the ring operated from the right. That is, the picture is now of the form f
and r, s are then the points (1) - (4) always written right of x, y.

ring is called commutative if the Multiplication in this ring is commutative.
If R is a commutative ring, so there is no difference between left and right module.
We only need the conditions for the R-module links verified.
If M is a left R-module over a commutative ring R, then M is called an R-module .

The following definitions can be there, where it says "module", always "left module" or always use "right module.

Let M be an R-module and N is a nonempty subset of N. Then N is a submodule means of M if N itself is an R-module. We write for "N is submodule of M"
Let N be a subset of a module M, it means the smallest submodule of M containing N, N of the product. We write for the product of N Let P be a subset of M. P is linearly dependent , if a finite number of pairwise different
If this is not the case, P is linearly independent .

Let B be a linearly independent subset of a module M with
Then B is a base M.

Be Rel is a relation on a set M.
Rel is reflexive applies if:
Rel is antisymmetric applies if:
Rel is transitive when:
A relation is partial order if it is reflexive, antisymmetric and transitive.
is For example, the set inclusion
a partial order.
says this: A is a subset of B.
A set with a partial order is called a partially ordered set .

Let R be a division ring and M a left R-module. The partially ordered by set inclusion set of all submodules of M is called the projective geometry P (M).

Let P (M) is a projective geometry. A flag of P (M) is a chain of pairwise different submodules

A real flag is a flag in which neither {0} or M appears.

One can interpret flags as sets of sub-modules. These we can sort by inclusion. The set of all real flags of a projective geometry, provided with the set inclusion, we call
A Frame in a projective geometry P (M) is a set of points
The Apartment S (F) of the frame F consists of the real flags
The set of all real flags and the set of all apartments in P (M)
give the building of P (M).