Episode space

A sequence space is a vector space considered in mathematics , the elements of which are sequences of numbers . Many vector spaces appearing in functional analysis are sequence spaces or can be represented by such. Examples include a. the important spaces like all bounded sequences or all sequences converging to zero. The sequence rooms offer a variety of possibilities for the construction of examples and can therefore also be viewed as a playground for functional analysts. ${\ displaystyle \ ell ^ {\ infty}}$${\ displaystyle c_ {0}}$

introduction

The vector space of all sequences in (= or ) is denoted by. Sequences can be added component by component and multiplied by real or complex numbers. Are about and such consequences and is , so is ${\ displaystyle \ omega}$${\ displaystyle \ mathbb {K}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle (x_ {n}) _ {n} = (x_ {1}, x_ {2}, x_ {3}, \ ldots)}$${\ displaystyle (y_ {n}) _ {n} = (y_ {1}, y_ {2}, y_ {3}, \ ldots)}$${\ displaystyle \ alpha \ in {\ mathbb {K}}}$

${\ displaystyle (x_ {n}) _ {n} + (y_ {n}) _ {n}: = (x_ {n} + y_ {n}) _ {n} = (x_ {1} + y_ { 1}, x_ {2} + y_ {2}, x_ {3} + y_ {3}, \ ldots)}$
${\ displaystyle \ alpha \ cdot (x_ {n}) _ {n}: = (\ alpha x_ {n}) _ {n} = (\ alpha x_ {1}, \ alpha x_ {2}, \ alpha x_ {3}, \ ldots)}$.

It is clear that with these operations there is a vector space. Sequence spaces are subspaces of this vector space which, in order to ensure a minimum richness , contain all sequences which are 1 in the -th position and 0 everywhere else. ${\ displaystyle \ omega}$${\ displaystyle \ mathbb {K}}$${\ displaystyle e ^ {(n)}}$${\ displaystyle n}$

The smallest sequence space is thus the subspace generated by the sequences . This is denoted by and consists of all sequences that only differ from 0 in a finite number of places. It is therefore also called the space of finite sequences, imagining that every finite sequence is continued by zeros to form an infinite sequence. So sequence spaces are subspaces of which contain. ${\ displaystyle e ^ {(n)}}$${\ displaystyle c_ {00}}$${\ displaystyle \ omega}$${\ displaystyle c_ {00}}$

The fact that the elements of a sequence space are sequences, which as elements of a vector space are also simply called points or vectors, can lead to misunderstandings. Especially when looking at sequences in such spaces, one is dealing with sequences of sequences.

In the following, norms or systems of norms or semi-norms are defined on sequence spaces. This gives normalized spaces or locally convex spaces .

c 0 and c

Probably the most well-known sequence spaces are the space of all sequences converging towards 0 and the space of all convergent sequences. If one looks at the supreme norm on these spaces , i. H. , so one obtains Banach spaces . The space is a subspace of the codimension 1. If the constant sequence, which equals 1 at every point, then applies . With multiplication explained by component, and Banach algebras , even C * algebras . It can further be shown that in is tight. Both spaces are thus separable , because the set of all finite sequences with values ​​from or is countable and dense. ${\ displaystyle c_ {0}}$${\ displaystyle c}$${\ displaystyle \ | (x_ {n}) _ {n} \ | _ {\ ell ^ {\ infty}}: = \ sup _ {n \ in {\ mathbb {N}}} | x_ {n} | }$${\ displaystyle c_ {0}}$${\ displaystyle c}$${\ displaystyle e}$${\ displaystyle c = c_ {0} \ oplus {\ mathbb {K}} \ cdot e}$${\ displaystyle c_ {0}}$${\ displaystyle c}$ ${\ displaystyle c_ {00}}$${\ displaystyle c_ {0}}$${\ displaystyle \ mathbb {Q}}$${\ displaystyle {\ mathbb {Q}} + i {\ mathbb {Q}}}$

ℓ p

It is the space of limited consequences with the supreme norm. For be ${\ displaystyle \ ell ^ {\ infty}}$${\ displaystyle 0

${\ displaystyle \ ell ^ {p}: = \ {(x_ {n}) _ {n} \ in \ omega; \, \ sum _ {n = 1} ^ {\ infty} | x_ {n} | ^ {p} <\ infty \}}$.

Is , the definition gives a metric that turns it into a complete topological vector space that is not a normalized space. For is through ${\ displaystyle 0 ${\ displaystyle \ textstyle d_ {p} ((x_ {n}) _ {n}, (y_ {n}) _ {n}): = \ sum _ {n = 1} ^ {\ infty} | x_ { n} -y_ {n} | ^ {p}}$${\ displaystyle \ ell ^ {p}}$${\ displaystyle 1 \ leq p <\ infty}$

${\ displaystyle \ | (x_ {n}) _ {n} \ | _ {\ ell ^ {p}}: = \ left (\ sum _ {n = 1} ^ {\ infty} | x_ {n} | ^ {p} \ right) ^ {\ frac {1} {p}}}$

defines the p -norm (for this one needs the Minkowski inequality ), which turns into a Banach space. The subspace is tight and the separability of for follows . The room is not separable. If namely , then let the sequence be 1 and 0 otherwise at every component . Then the uncountable number of sequences have in pairs the distance 1 from one another, which is why they cannot be separable. ${\ displaystyle \ ell ^ {p}}$${\ displaystyle c_ {00}}$${\ displaystyle \ ell ^ {p}}$${\ displaystyle p <\ infty}$${\ displaystyle \ ell ^ {\ infty}}$${\ displaystyle A \ subset \ mathbb {N}}$${\ displaystyle \ chi _ {A}}$${\ displaystyle A}$${\ displaystyle \ chi _ {A}}$${\ displaystyle \ ell ^ {\ infty}}$${\ displaystyle \ ell ^ {\ infty}}$

The -spaces are a special case of the more general L p -spaces if one considers the counting measure in the space . ${\ displaystyle \ ell ^ {p}}$${\ displaystyle \ mathbb {N}}$

Below the rooms is the Hilbert room ; according to Fischer-Riesz's theorem , apart from isometric isomorphism, this is the only infinite-dimensional separable Hilbert space. With the component-wise multiplication, all spaces are Banach algebras, is an H * algebra , a C * algebra, even a Von Neumann algebra . ${\ displaystyle \ ell ^ {p}}$ ${\ displaystyle \ ell ^ {2}}$${\ displaystyle \ ell ^ {p}}$${\ displaystyle \ ell ^ {2}}$${\ displaystyle \ ell ^ {\ infty}}$

duality

One says that the normalized sequence space has the normalized sequence space as a dual space if the following applies: ${\ displaystyle E}$${\ displaystyle F}$

1. For everyone and is .${\ displaystyle (x_ {n}) _ {n} \ in E}$${\ displaystyle (y_ {n}) _ {n} \ in F}$${\ displaystyle \ sum _ {n = 1} ^ {\ infty} | x_ {n} y_ {n} | <\ infty}$
2. Each defined by a continuous linear functional on .${\ displaystyle y = (y_ {n}) _ {n}}$${\ displaystyle \ phi _ {y} ((x_ {n}) _ {n}): = \ sum _ {n = 1} ^ {\ infty} x_ {n} y_ {n}}$ ${\ displaystyle E}$
3. The illustration is surjective and isometric .${\ displaystyle \ phi: F \ rightarrow E \, ', y \ mapsto \ phi _ {y}}$

In particular, since isometry implies injectivity, it is an isometric isomorphism. ${\ displaystyle \ phi}$

In this sense, the following dualities exist:

• ${\ displaystyle c_ {0} \, '= \ ell ^ {1}}$
• ${\ displaystyle \ ell ^ {1} \, '= \ ell ^ {\ infty}}$
• Is and so is .${\ displaystyle 1 ${\ displaystyle {\ tfrac {1} {p}} + {\ tfrac {1} {q}} = 1}$${\ displaystyle \ ell ^ {p} \, '= \ ell ^ {q}}$

Local convex spaces

Purely algebraically one has the isomorphies and . This means that the sum topology, i.e. the final topology of all inclusions , can be used to define what makes this space a (LF) space . is determined by the product topology, i. H. through the topology of component-wise convergence, to a locally convex space. ${\ displaystyle \ textstyle c_ {00} \ cong \ bigoplus _ {n = 1} ^ {\ infty} {\ mathbb {K}}}$${\ displaystyle \ textstyle \ omega \ cong \ prod _ {n = 1} ^ {\ infty} {\ mathbb {K}}}$${\ displaystyle c_ {00}}$${\ displaystyle {\ mathbb {K}} ^ {n} \ subset c_ {00}}$${\ displaystyle \ omega}$

The above-defined duality for standardized sequence spaces can be generalized to locally convex spaces if point 3 is replaced by the following requirement:

• The mapping is a homeomorphism .${\ displaystyle \ phi \ colon F \ rightarrow E \, ', y \ mapsto \ phi _ {y}}$

Then and . ${\ displaystyle c_ {00} '\, = \ omega}$${\ displaystyle \ omega \, '= c_ {00}}$

Köthe rooms

The following construction of locally convex sequence spaces, which can be traced back to Gottfried Köthe , offers a rich arsenal of examples.

A Köthe matrix is an infinite matrix with the following properties: ${\ displaystyle A = (a_ {n, m}) _ {n, m}}$

• ${\ displaystyle a_ {n, m} \ geq 0}$for all matrix elements and for each there is a with .${\ displaystyle n}$${\ displaystyle m}$${\ displaystyle a_ {n, m}> 0}$
• ${\ displaystyle a_ {n, m} \ leq a_ {n, m + 1}}$for all indices .${\ displaystyle n, m}$

With this data the following spaces are defined, where : ${\ displaystyle 1 \ leq p <\ infty}$

${\ displaystyle \ lambda ^ {p} (A): = \ {(x_ {n}) _ {n} \ in \ omega: \ | (x_ {n}) _ {n} \ | _ {m}: = (\ sum _ {n = 1} ^ {\ infty} | x_ {n} \ cdot a_ {n, m} | ^ {p}) ^ {\ frac {1} {p}} <\ infty \, \, \ forall m \ in {\ mathbb {N}} \}}$

${\ displaystyle \ lambda ^ {\ infty} (A): = \ {(x_ {n}) _ {n} \ in \ omega: \ | (x_ {n}) _ {n} \ | _ {m} : = \ sup _ {n \ in {\ mathbb {N}}} | x_ {n} | \ cdot a_ {n, m} <\ infty \, \, \ forall m \ in {\ mathbb {N}} \}}$

${\ displaystyle c_ {0} (A): = \ {(x_ {n}) _ {n} \ in \ lambda ^ {\ infty}: \ lim _ {n \ to \ infty} | x_ {n} | \ cdot a_ {n, m} = 0 \, \, \ forall m \ in {\ mathbb {N}} \}}$.

These spaces are called the Köthe spaces defined by the Köthe matrix (or Koethesche step spaces ), the norms are called the associated canonical norms . With the system of canonical norms, each of these spaces becomes a locally convex space, even a Fréchet space . ${\ displaystyle \ | \ cdot \ | _ {m}}$

If the Köthe matrix is ​​selected as the matrix that is equal to 1 at each component, the normalized spaces defined above are returned: , . By choosing Köthe matrices, the matrix elements of which show a certain growth behavior, you can construct examples for completely different room classes. ${\ displaystyle I}$${\ displaystyle \ lambda ^ {p} (I) = \ ell ^ {p}}$${\ displaystyle c_ {0} (I) = c_ {0}}$

So z. B .:

The following statements are equivalent for a Köthe matrix : ${\ displaystyle A = (a_ {n, m}) _ {n, m}}$

• There is a Montel room for each .${\ displaystyle p \ in [1, \ infty]}$${\ displaystyle \ lambda ^ {p} (A)}$
• ${\ displaystyle c_ {0} (A)}$ is a Montel room.
• For each infinite subset and each there is one such that .${\ displaystyle N \ subset \ mathbb {N}}$${\ displaystyle m \ in \ mathbb {N}}$${\ displaystyle k \ in \ mathbb {N}}$${\ displaystyle \ inf _ {n \ in N} {\ frac {a_ {n, m}} {a_ {n, k}}} = 0}$

The following statements are equivalent for a Köthe matrix : ${\ displaystyle A = (a_ {n, m}) _ {n, m}}$

• There is a Schwartz room for each .${\ displaystyle p \ in [1, \ infty]}$${\ displaystyle \ lambda ^ {p} (A)}$
• For each there is one , so that .${\ displaystyle m \ in \ mathbb {N}}$${\ displaystyle k \ geq m}$${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {a_ {n, m}} {a_ {n, k}}} = 0}$

The following statements are equivalent for a Köthe matrix : ${\ displaystyle A = (a_ {n, m}) _ {n, m}}$

• For each there is a nuclear space .${\ displaystyle p \ in [1, \ infty]}$${\ displaystyle \ lambda ^ {p} (A)}$
• ${\ displaystyle c_ {0} (A)}$ is a nuclear space.
• For each there is one , so that .${\ displaystyle m \ in \ mathbb {N}}$${\ displaystyle k \ geq m}$${\ displaystyle \ sum _ {n = 1} ^ {\ infty} {\ frac {a_ {n, m}} {a_ {n, k}}} <\ infty}$

As an application of these statements, one can construct examples of Montel spaces that are not Schwartz spaces by choosing a suitable Köthe matrix. Such examples are very important to bring some order to the zoo of locally convex spaces .

For the matrix one calls the space of the rapidly falling sequences . This space plays an important role in the theory of nuclear spaces, because according to the Kōmura-Kōmura theorem , this space is a generator of all nuclear spaces. ${\ displaystyle A = (n ^ {m}) _ {n, m}}$${\ displaystyle s: = \ lambda ^ {1} (A)}$${\ displaystyle s}$