Nakayama lemma

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L² of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

For many authors, the Riesz–Fischer theorem refers to the fact that the L^p spaces from Lebesgue integration theory are complete.

Modern forms of the theorem

The most common form of the theorem states that a measurable function on [–π, π] is square integrable if and only if the corresponding Fourier series converges in the space L². This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by

$S_{N}f(x)=\sum _{n=-N}^N{F}_n{\mathrm{e}}^{inx},$

where F_n, the nth Fourier coefficient, is given by

$F_n=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}f(x){\mathrm{e}}^{-inx}\mathrm{d}x,$

then

$\underset{N\to \mathrm{\infty }}{\lim }{\Vert S_{N}f-f\Vert }_2=0,$

where ${\Vert \cdot \Vert }_2$ is the L²-norm.

Conversely, if $\left\{a_n\right\}$ is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left|a_n\right|}^2<\mathrm{\infty },$

then there exists a function f such that f is square-integrable and the values $a_n$ are the Fourier coefficients of f.

This form of the Riesz–Fischer theorem is a stronger form of Bessel's inequality, and can be used to prove Parseval's identity for Fourier series.

Other results are often called the Riesz–Fischer theorem Template:Harv. Among them is the theorem that, if A is an orthonormal set in a Hilbert space H, and x ∈ H, then

$⟨x,y⟩=0$

for all but countably many y ∈ A, and

$\sum _{y\in A}|⟨x,y⟩{|}^2\le \Vert x{\Vert }^2.$

Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series

$\sum _{y\in A}⟨x,y⟩y$

converges commutatively (or unconditionally) to x. This is equivalent to saying that for every ε > 0, there exists a finite set B₀ in A such that

$\Vert x-\sum _{y\in B}⟨x,y⟩y\Vert <\epsilon$

for every finite set B containing B₀. Moreover, the following conditions on the set A are equivalent:

• the set A is an orthonormal basis of H
• for every vector x ∈ H,

$\Vert x{\Vert }^2=\sum _{y\in A}|⟨x,y⟩{|}^2.$

Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that L² (or more generally L^p, 0 < p ≤ ∞) is complete.

Example

The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let {${\varphi }_n$} be an orthonormal system in R (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. – see orthogonal polynomials), not necessarily complete (in an inner product space, an orthonormal set is complete if no nonzero vector is orthogonal to every vector in the set). The theorem asserts that if the normed space R is complete (thus R is a Hilbert space), then any sequence {$c_n$} that has finite ℓ² norm defines a function f in the space R.

The function f is defined by $f=\underset{n\to \mathrm{\infty }}{\lim }\sum _{k=0}^n{c}_k{\varphi }_k$, limit in R-norm.

Combined with the Bessel's inequality, we know the converse as well: if f is a function in R, then the Fourier coefficients $(f,{\varphi }_n)$ have finite ℓ² norm.

History: the Note of Riesz and the Note of Fischer (1907)

In his Note, Template:Harvtxt states the following result (translated here to modern language at one point: the notation L²([a, b]) was not used in 1907).

Let {φ_n } be an orthonormal system in L²([a, b]) and {a_n } a sequence of reals. The convergence of the series $\sum a_n^2$ is a necessary and sufficient condition for the existence of a function f such that

$\underset{a}{\overset{b}{\int }}f(x){\phi }_n(x)\mathrm{d}x=a_n$

for every n.

Today, this result of Riesz is a special case of basic facts about series of orthogonal vectors in Hilbert spaces.

Riesz's Note appeared in March. In May, Template:Harvtxt states explicitly in a theorem (almost with modern words) that a Cauchy sequence in L²([a, b]) converges in L²-norm to some function f  in L²([a, b]). In this Note, Cauchy sequences are called "sequences converging in the mean" and L²([a, b]) is denoted by Ω. Also, convergence to a limit in L²–norm is called "convergence in the mean towards a function". Here is the statement, translated from French:

Theorem. If a sequence of functions belonging to Ω  converges in the mean, there exists in Ω a function f towards which the sequence converges in the mean.

Fischer goes on proving the preceding result of Riesz, as a consequence of the orthogonality of the system, and of the completeness of L².

Fischer's proof of completeness is somewhat indirect. It uses the fact that the indefinite integrals of the functions g_n in the given Cauchy sequence, namely

$G_n(x)=\underset{a}{\overset{x}{\int }}{g}_n(t)\mathrm{d}t,$

converge uniformly on [a, b] to some function G, continuous with bounded variation. The existence of the limit g ∈ L² for the Cauchy sequence is obtained by applying to G differentiation theorems from Lebesgue's theory.
Riesz uses a similar reasoning in his Note, but makes no explicit mention to the completeness of L², although his result may be interpreted this way. He says that integrating term by term a trigonometric series with given square summable coefficients, he gets a series converging uniformly to a continuous function F  with bounded variation. The derivative f  of F, defined almost everywhere, is square summable and has for Fourier coefficients the given coefficients.

Completeness of L^p,  0 < p ≤ ∞

The proof that L^p is complete is based on the convergence theorems for the Lebesgue integral.

When 1 ≤ p ≤ ∞, the Minkowski inequality implies that the space L^p is a normed space. In order to prove that L^p is complete, i.e. that L^p is a Banach space, it is enough (see e.g. Banach_space#Definition) to prove that every series ∑ u_n of functions in L^p(μ) such that

$\sum \Vert u_n{\Vert }_p<\mathrm{\infty }$

converges in the L^p-norm to some function f ∈ L^p(μ). For p < ∞, the Minkowski inequality and the monotone convergence theorem imply that

$\int \left(\sum _{n=0}^{\mathrm{\infty }}\right|u_n|{)}^p\mathrm{d}\mu \le (\sum _{n=0}^{\mathrm{\infty }}\Vert u_n{\Vert }_p{)}^p<\mathrm{\infty },\text{ hence }f=\sum _{n=0}^{\mathrm{\infty }}{u}_n$

is defined μ–almost everywhere and f ∈ L^p(μ). The dominated convergence theorem is then used to prove that the partial sums of the series converge to f in the L^p-norm,

$\int {|f-\sum _{k=0}^n{u}_k|}^p\mathrm{d}\mu \le \int {\left(\sum _{\ell >n}\right|u_{\ell }\left|\right)}^p\mathrm{d}\mu \to 0\text{ as }n\to \mathrm{\infty }.$

The case 0 < p < 1 requires some modifications, due to the fact that the p-norm is no longer subadditive. One starts with the stronger assumption that

$\sum \Vert u_n{\Vert }_p^p<\mathrm{\infty }$

and uses repeatedly that

${\left|\sum _{k=0}^n{u}_k\right|}^p\le \sum _{k=0}^n|u_k{|}^p\text{ when }p<1$

The case p = ∞ reduces to a simple question about uniform convergence outside a μ-negligible set.

References

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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.