Introduction into Hilbert Spaces

Abstract: This is lecture notes for the course Hilbert Spaces at School of Mathematics of University of Leeds. They are based on the notes of Prof. Jonathan R. Partington used in the previous years. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more.

The notes are available also for download in PDF.

The suggested textbooks are [1, 2, 3, 4]. The other nice books with many interesting problems are [5, 6].

Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them!

Contents

• 1  Basics of Linear Spaces
  • 1.1  Banach spaces (basic definitions only)
  • 1.2  Hilbert spaces
  • 1.3  Subspaces
  • 1.4  Linear spans
• 2  Orthogonality
  • 2.1  Orthogonal System in Hilbert Space
  • 2.2  Bessel’s inequality
  • 2.3  The Riesz–Fischer theorem
  • 2.4  Construction of Orthonormal Sequences
  • 2.5  Orthogonal complements
• 3  Fourier Analysis
  • 3.1  Fourier series
• 4  Fejér’s theorem
  • 4.1  Parseval’s formula
  • 4.2  Some Application of Fourier Series
• 5  Duality of Linear Spaces
  • 5.1  Dual space of a normed space
  • 5.2  Self-duality of Hilbert space
• 6  Operators
  • 6.1  Linear operators
  • 6.2  B(H) as a Banach space (and even algebra)
  • 6.3  Adjoints
  • 6.4  Hermitian, unitary and normal operators
• 7  Spectral Theory
  • 7.1  The spectrum of an operator on a Hilbert space
  • 7.2  The spectral radius formula
  • 7.3  Spectrum of Special Operators
• 8  Compactness
  • 8.1  Compact operators
  • 8.2  Hilbert–Schmidt operators
• 9  Compact normal operators
  • 9.1  Spectrum of normal operators
  • 9.2  Compact normal operators
• 10  Integral equations
• A  Tutorial Problems
  • A.1  Tutorial problems I
  • A.2  Tutorial problems II
  • A.3  Tutorial Problems III
  • A.4  Tutorial Problems IV
  • A.5  Tutorial Problems V
  • A.6  Tutorial Problems VI
  • A.7  Tutorial Problems VII
• B  Solutions of Tutorial Problems
  • B.1  Solution of Tuitorial Problem I
  • B.2  Solutions of Tutorial Problems II
  • B.3  Solutions of Tutorial Problems III
  • B.4  Solutions of Tutorial Problems IV
  • B.5  Solutions of Tutorial Problems V
  • B.6  Solutions of Tutorial Problems VI
  • B.7  Solutions of Tutorial Problems VII
• C  Course in the Nutshell
  • C.1  Some useful results and formulae (1)
  • C.2  Some useful results and formulae (2)
  • C.3  Examinable material
• D  Supplementary Sections
  • D.1  Reminder from Complex Analysis

Notations

ℤ₊, ℝ₊ denotes non-negative integers and reals.
x,y,z,… denotes vectors.
λ,µ,ν,… denotes scalars.
ℜ z, ℑ z stand for real and imaginary parts of a complex number z.

1  Basics of Linear Spaces

    A person is solely the concentration of an infinite set of interrelations with another and others, and to separate a person from these relations means to take away any real meaning of the life.

  Vl. Soloviev

A space around us could be described as a three dimensional Euclidean space. To single out a point of that space we need a fixed frame of references and three real numbers, which are coordinates of the point. Similarly to describe a pair of points from our space we could use six coordinates; for three points—nine, end so on. This makes it reasonable to consider Euclidean (linear) spaces of an arbitrary finite dimension, which are studied in the courses of linear algebra.

The basic properties of Euclidean spaces are determined by its linear and metric structures. The linear space (or vector space) structure allows to add and subtract vectors associated to points as well as to multiply vectors by real or complex numbers (scalars).

The metric space structure assign a distance—non-negative real number—to a pair of points or, equivalently, defines a length of a vector defined by that pair. A metric (or, more generally a topology) is essential for definition of the core analytical notions like limit or continuity. The importance of linear and metric (topological) structure in analysis sometime encoded in the formula:

On the other hand we could observe that many sets admit a sort of linear and metric structures which are linked each other. Just few among many other examples are:

• The set of convergent sequences;
• The set of continuous functions on [0,1].

It is a very mathematical way of thinking to declare such sets to be spaces and call their elements points.

But shall we lose all information on a particular element (e.g. a sequence {1/n}) if we represent it by a shapeless and size-less “point” without any inner configuration? Surprisingly not: all properties of an element could be now retrieved not from its inner configuration but from interactions with other elements through linear and metric structures. Such a “sociological” approach to all kind of mathematical objects was codified in the abstract category theory.

Another surprise is that starting from our three dimensional Euclidean space and walking far away by a road of abstraction to infinite dimensional Hilbert spaces we are arriving just to yet another picture of the surrounding space—that time on the language of quantum mechanics.

1.1  Banach spaces (basic definitions only)

    The distance from Manchester to Liverpool is 35 miles—just about the mileage in the opposite direction!

   A tourist guide to UK

The following definition generalises the notion of distance known from the everyday life.

The following notion is a useful specialisation of metric adopted to the linear structure.

The connection between norm and metric is as follows:

Proof.

---

    (a)       (b)       

Figure 1: Triangle inequality in metric (a) and normed (b) spaces.

---

This is a simple exercise to derive items 1–3 of Definition 1 from corresponding items of Definition 3. For example, see the Figure 1 to derive the triangle inequality.

An important notions known from real analysis are limit and convergence. Particularly we usually wish to have enough limiting points for all “reasonable” sequences.

For example, the set of integers ℤ and reals ℝ with the natural distance functions are complete spaces, but the set of rationals ℚ is not. The complete normed spaces deserve a special name.

---

(i)       (ii)      (iii)

Figure 2: Different unit balls defining norms in ℝ2 from Example 8.

---

1.2  Hilbert spaces

    —We need an extra space to accommodate this product!

   A manager to a shop assistant

Although metric and norm capture important geometric information about linear spaces they are not sensitive enough to represent such geometric characterisation as angles (particularly orthogonality). To this end we need a further refinements.

From courses of linear algebra known that the scalar product ⟨ x,y ⟩= x₁ y₁ + ⋯ + x_n y_n is important in a space ℝⁿ and defines a norm ||x||²=⟨ x,x ⟩. Here is a suitable generalisation:

Last two properties of the scalar product is oftenly encoded in the phrase: “it is linear in the first variable if we fix the second and anti-linear in the second if we fix the first”.

Now we state, probably, the most important inequality in analysis.

Proof. If x=λ y then it is easy. Otherwise x−λ y≠ 0 for any λ, particularly for λ=⟨ x,y ⟩/⟨ y,y ⟩ we have:

which yields our goal.

Proof. Just to check items 1–3 from Definition 3.

Again complete inner product spaces deserve a special name

The relations between spaces introduced so far are as follows:

How can we tell if a given norm comes from an inner product?

---

Figure 3: To the parallelogram identity.

---

Proof. Just by linearity of inner product:

because the cross terms cancel out.

1.3  Subspaces

    Divide and rule!

   Old but still much used recipe

To study Hilbert spaces we may use the traditional mathematical technique of analysis and synthesis: we split the initial Hilbert spaces into smaller and probably simpler subsets, investigate them separately, and then reconstruct the entire picture from these parts.

As known from the linear algebra, a linear subspace is a subset of a linear space is its subset, which inherits the linear structure, i.e. possibility to add vectors and multiply them by scalars. In this course we need also that subspaces inherit topological structure (coming either from a norm or an inner product) as well.

We also wish that the both inhered structures (linear and topological) should be in agreement, i.e. the subspace should be complete. Such inheritance is linked to the property be closed.

A subspace need not be closed—for example the sequence

and x_n=(1, 1/2,…, 1/n, 0, 0,…)∈ c₀₀ converges to x thus x∈ c₀₀ ⊂ l₂.

Proof.

1. This is true in any metric space X: any Cauchy sequence from Y has a limit x ∈ X belonging to Ȳ, but if Y is closed then x ∈ Y.
2. Let Y is complete and x∈ Ȳ, then there is sequence x_n→ x in Y and it is a Cauchy sequence. Then completeness of Y implies x∈ Y.
3. If x, y∈ Ȳ then there are x_n and y_n in Y such that x_n→ x and y_n→ y. From the  triangle inequality:

   ⎪⎪ ⎪⎪

   (xn+yn)−(x+y)

   ⎪⎪ ⎪⎪

   ≤

   ⎪⎪ ⎪⎪

   xn−x

   ⎪⎪ ⎪⎪

   +

   ⎪⎪ ⎪⎪

   yn−y

   ⎪⎪ ⎪⎪

   → 0,

   so x_n+y_n→ x+y and x+y∈ Ȳ. Similarly x∈Ȳ implies λ x ∈Ȳ for any λ.

Hence c₀₀ is an incomplete inner product space, with inner product ⟨ x,y ⟩=∑₁^∞x_k ȳ_k (this is a finite sum!) as it is not closed in l₂.

---

(a)          (b)  

Figure 4: Jump function on (b) as a L2 limit of continuous functions from (a).

---

Similarly C[0,1] with inner product norm ||f||=(∫₀¹ | f(t) |² dt)^1/2 is incomplete—take the large space X of functions continuous on [0,1] except for a possible jump at 1/2 (i.e. left and right limits exists but may be unequal and f(1/2)=lim_t→1/2+ f(t). Then the sequence of functions defined on Figure 4(a) has the limit shown on Figure 4(b) since:

Obviously f∈C[0,1]∖C[0,1].

Similarly the space C[a,b] is incomplete for any a<b if equipped by the inner product and the corresponding norm:

It is practical to realise L₂[a,b] as a certain space of “functions” with the inner product defined via an integral. There are several ways to do that and we mention just two:

1. Elements of L₂[a,b] are equivalent classes of Cauchy sequences f⁽ⁿ⁾ of functions from C[a,b].
2. Let integration be extended from the Riemann definition to the wider Lebesgue integration (see MATH4011). Let L be a set of square integrable in Lebesgue sense functions on [a,b] with a finite norm (8). Then L₂[a,b] is a quotient space of L with respect to the equivalence relation f∼ g ⇔ ||f−g||₂=0 .
   Example 22   Let the Cantor function on [0,1] be defined as follows:

   f(t)=

   ⎧ ⎨ ⎩

   1, t∈ ℚ;           0, t∈ ℝ∖ℚ.

   This function is not integrable in the Riemann sense but does have the Lebesgue integral. The later however is equal to 0 and as an L₂-function the Cantor function equivalent to the function identically equal to 0.
3. The third possibility is to map L₂(ℝ) onto a space of “true” functions but with an additional structure. For example, in quantum mechanics it is useful to work with the Segal–Bargmann space of analytic functions on ℂ with the inner product:

   ⟨ f1,f2  ⟩=

   ∫

   ℂ

   f1(z) f2(z)e

   − ⎪ ⎪ z  ⎪ ⎪ 2

   dz.

Proof. Take a Cauchy sequence x⁽ⁿ⁾∈l₂, where x⁽ⁿ⁾=(x₁⁽ⁿ⁾, x₂⁽ⁿ⁾, x₃⁽ⁿ⁾, … ). Our proof will have three steps: identify the limit x; show it is in l₂; show x⁽ⁿ⁾→ x.

1. If x⁽ⁿ⁾ is a Cauchy sequence in l₂ then x_k⁽ⁿ⁾ is also a Cauchy sequence of numbers for any fixed k:

   ⎪ ⎪

   xk(n)−xk(m)

   ⎪ ⎪

   ≤

   ⎛ ⎜ ⎜ ⎝

   ∞ ∑ k=1

   ⎪ ⎪

   xk(n)−xk(m)

   ⎪ ⎪

   2

   ⎞ ⎟ ⎟ ⎠

   1/2

   =

   ⎪⎪ ⎪⎪

   x(n)−x(m)

   ⎪⎪ ⎪⎪

   → 0.

   Let x_k be the limit of x_k⁽ⁿ⁾.
2. For a given є>0 find n₀ such that ||x⁽ⁿ⁾−x^(m)||<є for all n,m>n₀. For any K and m:

   K ∑ k=1

   ⎪ ⎪

   xk(n)−xk(m)

   ⎪ ⎪

   2 ≤

   ⎪⎪ ⎪⎪

   x(n)−x(m)

   ⎪⎪ ⎪⎪

   2<є2.

   Let m→ ∞ then ∑_k=1^K | x_k⁽ⁿ⁾−x_k |² ≤ є².
   Let K→ ∞ then ∑_k=1^∞| x_k⁽ⁿ⁾−x_k |² ≤ є². Thus x⁽ⁿ⁾−x∈l₂ and because l₂ is a linear space then x = x⁽ⁿ⁾−(x⁽ⁿ⁾−x) is also in l₂.
3. We saw above that for any є >0 there is n₀ such that ||x⁽ⁿ⁾−x||<є for all n>n₀. Thus x⁽ⁿ⁾→ x.

Consequently l₂ is complete.

1.4  Linear spans

    All good things are covered by a thick layer of chocolate (well, if something is not yet–it certainly will)

As was explained into introduction 1, we describe “internal” properties of a vector through its relations to other vectors. For a detailed description we need sufficiently many external reference points.

Let A be a subset (finite or infinite) of a normed space V. We may wish to upgrade it to a linear subspace in order to make it subject to our theory.

Proof. Clearly Lin(A) is a closed subspace containing A thus it should contain CLin(A). Also Lin(A)⊂ CLin(A) thus Lin(A)⊂ CLin(A)=CLin(A). Therefore Lin(A)= CLin(A).

Consequently CLin(A) is the set of all limiting points of finite linear combination of elements of A.

The following simple result will be used later many times without comments.

Proof. Obviously by the Cauchy–Schwarz inequality:

since ||x_n−x||→ 0, ||y_n−y||→ 0, and ||y_n|| is bounded.

2  Orthogonality

    Pythagoras is forever!

  The catchphrase from TV commercial of Hilbert Spaces course

As was mentioned in the introduction the Hilbert spaces is an analog of our 3D Euclidean space and theory of Hilbert spaces similar to plane or space geometry. One of the primary result of Euclidean geometry which still survives in high school curriculum despite its continuous nasty de-geometrisation is Pythagoras’ theorem based on the notion of orthogonality¹.

So far we was concerned only with distances between points. Now we would like to study angles between vectors and notably right angles. Pythagoras’ theorem states that if the angle C in a triangle is right then c²=a²+b², see Figure 5 .

---

Figure 5: The Pythagoras’ theorem c2=a2+b2

---

It is a very mathematical way of thinking to turn this property of right angles into their definition, which will work even in infinite dimensional Hilbert spaces.

2.1  Orthogonal System in Hilbert Space

    Look for a triangle, or even for a right triangle

  A universal advice in solving problems from elementary geometry.

In inner product spaces it is even more convenient to give a definition of orthogonality not from Pythagoras’ theorem but from an equivalent property of inner product.

Proof. A one-line calculation.

The following theorem provides an important property of Hilbert spaces which will be used many times. Recall, that a subset K of a linear space V is convex if for all x, y∈ K and λ∈ [0,1] the point λ x +(1−λ)y is also in K. Particularly any subspace is convex and any unit ball as well (see Exercise 1).

Proof. Let d=inf_{y∈ K} d(x,y), where d(x,y)—the distance coming from the norm ||x||=√⟨ x,x ⟩ and let y_n a sequence points in K such that lim_{n→ ∞}d(x,y_n)=d. Then y_n is a Cauchy sequence. Indeed from the parallelogram identity for the parallelogram generated by vectors x−y_n and x−y_m we have:

Note that ||2x−y_n−y_m||²=4||x−y_n+y_m/2||²≥ 4d² since y_n+y_m/2∈ K by its convexity. For sufficiently large m and n we get ||x−y_m||²≤ d +є and ||x−y_n||²≤ d +є, thus ||y_n−y_m||≤ 4(d²+є)−4d²=4є, i.e. y_n is a Cauchy sequence.

Let y be the limit of y_n, which exists by the completeness of H, then y∈ K since K is closed. Then d(x,y)=lim_{n→ ∞}d(x,y_n)=d. This show the existence of the nearest point. Let y′ be another point in K such that d(x,y′)=d, then the parallelogram identity implies:

This shows the uniqueness of the nearest point.

2.2  Bessel’s inequality

    Liberte, Egalite, Fraternite!

  A longstanding ideal approximated in the real life by something completely different

For the case then a convex subset is a subspace we could characterise the nearest point in the term of orthogonality.

Proof. Let z is the nearest point to x existing by the previous Theorem. We claim that x−z orthogonal to any vector in M, otherwise there exists y∈ M such that ⟨ x−z,y ⟩≠ 0. Then

if є is chosen to be small enough and such that є ℜ⟨ x−z,y ⟩ is positive, see Figure 6(i). Therefore we get a contradiction with the statement that z is closest point to x.

On the other hand if x−z is orthogonal to all vectors in H₁ then particularly (x−z)⊥ (z−y) for all y∈ H₁, see Figure 6(ii). Since x−y=(x−z)+(z−y) we got by the Pythagoras’ theorem:

---

    (i)     (ii)    

Figure 6: (i) A smaller distance for a non-perpendicular direction; and

(ii) Best approximation from a subspace

---

So ||x−y||²≥ ||x−z||² and the are equal if and only if z=y.

Consider now a basic case of approximation: let x∈ H be fixed and e₁, …, e_n be orthonormal and denote H₁=Lin{e₁,…,e_n}. We could try to approximate x by a vector y=λ₁ e₁+⋯ +λ_n e_n ∈ H₁.

Proof. Let z=∑₁ⁿ⟨ x,e_i ⟩ e_i, then ⟨ x−z,e_i ⟩=⟨ x,e_i ⟩−⟨ z,e_i ⟩=0. By the previous Theorem z is the nearest point to x.

Proof. Let z= ∑₁ⁿ⟨ x,e_i ⟩e_i then x−z⊥ e_i for all i therefore by Exercise 4 x−z⊥ z. Hence:

2.3  The Riesz–Fischer theorem

    —Did you say “rice and fish for them”?

  A student question

When (e_i) is orthonormal we call ⟨ x,e_n ⟩ the nth Fourier coefficient of x (with respect to (e_i), naturally).

Proof. Necessity: Let x_k=∑₁^k λ_n e_n and x=lim_{k→ ∞} x_k. So ⟨ x,e_n ⟩=lim_{k→ ∞}⟨ x_k,e_n ⟩=λ_n for all n. By the Bessel’s inequality for all k

hence ∑₁^k | λ_n |² converges and the sum is at most ||x||².

Sufficiency: Consider ||x_k−x_m||=||∑_m^k λ_n e_n||=(∑_m^k | λ_n |²)^1/2 for k>m. Since ∑_m^k | λ_n |² converges x_k is a Cauchy sequence in H and thus has a limit x. By the Pythagoras’ theorem ||x_k||²=∑₁^k | λ_n |² thus for k→ ∞ ||x||²=∑₁^∞| λ_n |² by the Lemma about inner product limit.

Observation: the closed linear span of an orthonormal sequence in any Hilbert space looks like l₂, i.e. l₂ is a universal model for a Hilbert space.

By Bessel’s inequality and the Riesz–Fisher theorem we know that the series ∑₁^∞⟨ x,e_i ⟩ e_i converges for any x∈ H. What is its limit?

Let y=x− ∑₁^∞⟨ x,e_i ⟩ e_i, then

Proof. By the Riesz–Fisher theorem, equation (10) and definition of orthonormal basis.

2.4  Construction of Orthonormal Sequences

    There are constructive existence theorems in mathematics.

  An example of pure existence statement

Natural questions are: Do orthonormal sequences always exist? Could we construct them?

Proof. We give an explicit algorithm working by induction. The base of induction: the first vector is e₁=x₁/||x₁||. The step of induction: let e₁, e₂, …, e_n are already constructed as required. Let y_n+1=x_n+1−∑_i=1ⁿ⟨ x_n+1,e_i ⟩e_i. Then by (10) y_n+1 ⊥ e_i for i=1,…,n. We may put e_n+1=y_n+1/||y_n+1|| because y_n+1≠ 0 due to linear independence of x_k’s. Also

So (e_i) are orthonormal sequence.

Another natural question is: When is an orthonormal sequence complete?

Proof. Clearly 1 implies 2 because x=∑₁^∞⟨ x,e_n ⟩e_n in CLin((e_n)) and ||x||²=∑₁^∞⟨ x,e_n ⟩e_n by Theorem 15.

If (e_n) is not complete then there exists x∈ H such that x≠ 0 and ⟨ x,e_k ⟩ for all k, so 3 fails, consequently 3 implies 1.

Finally if ⟨ x,e_k ⟩=0 for all k then ⟨ x,y ⟩=0 for all y∈Lin((e_n)) and moreover for all y∈CLin((e_n)), by the Lemma on continuity of the inner product. But then x∉CLin((e_n)) and 2 also fails because ⟨ x,x ⟩=0 is not possible. Thus 2 implies 1.

Proof. Take a countable dense set (x_k), then H=CLin((x_k)), delete all vectors which are a linear combinations of preceding vectors, make orthonormalisation by Gram–Schmidt the remaining set and apply the previous proposition.

2.5  Orthogonal complements

    Most pleasant compliments are usually orthogonal to our real qualities.

  An advise based on observations

Proof. Clearly M^⊥ is a subspace of H because x, y∈ M^⊥ implies ax+by∈ M^⊥:

Also if all x_n∈ M^⊥ and x_n→ x then x∈ M^⊥ due to inner product limit Lemma.

Proof. For a given x there exists the unique closest point m in M by the Theorem on nearest point and by the Theorem on perpendicular (x−m)⊥ y for all y∈ M.

So x= m + (x−m)= m+n with m∈ M and n∈ M^⊥. The identity ||x||²=||m||²+||n||² is just Pythagoras’ theorem and M∩ M^⊥={0} because null vector is the only vector orthogonal to itself.

Finally (M^⊥)^⊥=M. We have H=M⊕ M^⊥=(M^⊥)^⊥⊕ M^⊥, for any x∈(M^⊥)^⊥ there is a decomposition x=m+n with m∈ M and n∈ M^⊥, but then n is orthogonal to itself and therefore is zero.

Proof. Let us define P_M(x)=m where x=m+n is the decomposition from the previous theorem. The linearity of this operator follows from the fact that both M and M^⊥ are linear subspaces. Also P_M(m)=m for all m∈ M and the image of P_M is M. Thus P_M²=P_M. Also if P_M(x)=0 then x⊥ M, i.e. kerP_M=M^⊥. Similarly P_M^⊥(x)=n where x=m+n and P_M+P_M^⊥=I.

3  Fourier Analysis

    All bases are equal, but some are more equal then others.

As we saw already any separable Hilbert space posses an orthonormal basis (infinitely many of them indeed). Are they equally good? This depends from our purposes. For solution of differential equation which arose in mathematical physics (wave, heat, Laplace equations, etc.) there is a proffered choice. The fundamental formula: d/dx e^ax=ae^ax reduces the derivative to a multiplication by a. We could benefit from this observation if the orthonormal basis will be constructed out of exponents.

3.1  Fourier series

    7.40pm Fourier series: Episode I

  TV listing for today

Let us consider the space L₂[−π,π]. As we saw in Example 3 there is an orthonormal sequence e_n(t)=(2π)^−1/2e^int in L₂[−π,π]. We will show that it is an orthonormal basis, i.e.

with convergence in L₂ norm. To do this we show that CLin{e_k:k∈ℤ}=L₂[−π,π].

Let CP[−π,π] denote the continuous functions f on [−π,π] such that f(π)=f(−π). We also define f outside of the interval [−π,π] by periodicity.

Proof. Let f∈L₂[−π,π]. Given є>0 there exists g∈ C[−π,π] such that ||f−g||<є/2. Form continuity of g on a compact set follows that there is M such that | g(t) |<M for all t∈[−π,π].

---

Figure 9: A modification of continuous function to periodic

---

We can now replace g by periodic g′, which coincides with g on [−π,π−δ] for an arbitrary δ>0 and has the same bounds: | g′(t) |<M, see Figure 9. Then

So if δ<є²/(4M)² then ||g−g′||<є/2 and ||f−g′||<є.

Now if we could show that CLin{e_k: k ∈ ℤ} includes CP[−π,π] then it also includes L₂[−π,π].

We want to show that ||f−f_n||₂→ 0. To this end we define nth Fejér sum by the formula

and show that

Then we conclude

Since F_n∈Lin((e_n)) then f∈CLin((e_n)) and hence f=∑_−∞^∞⟨ f,e_k ⟩e_k.

4  Fejér’s theorem

Proof. From notation (14):

Then from (15):

which finishes the proof.

Proof. Let z=e^it, then:

by switch from counting in rows to counting in columns in Table 1.

---

			1
		z−1	1	z
	z−2	z−1	1	z	z2
	⋮	⋮	⋮	⋮	⋮	⋱

Table 1: Counting powers in rows and columns

---

Let w=e^it/2, i.e. z=w², then

if w≠ ± 1. For the value of K_n(0) we substitute w=1 into (19).

---

Figure 10: You could see a family of Fejér kernels with the parameter m running from 0 to 10.

---

The first eleven Fejér kernels are shown on Figure 10, we could observe that:

Proof. The first property immediately follows from the explicit formula (18). In contrast the second property is easier to deduce from expression with double sum (17):

since the formula (9).

Finally if | t |>δ then sin²(t/2)≥ sin²(δ/2)>0 by monotonicity of sinus on [0,π/2], so:

implying:

Therefore the third property follows from the squeeze rule.

Proof. Idea of the proof: if in the formula (16)

t is long way from x, K_n is small (see Lemma 3 and Figure 10), for t near x, K_n is big with total “weight” 2π, so the weighted average of f(t) is near f(x).

Here are details. Using property 2 and periodicity of f and K_n we could express trivially

Similarly we rewrite (16) as

then

Given є>0 split into three intervals: I₁=[x−π,x−δ], I₂=[x−δ,x+δ], I₃=[x+δ,x+π], where δ is chosen such that | f(t)−f(x) |<є/2 for t∈ I₂, which is possible by continuity of f. So

And

if n is sufficiently large due to property 3 of K_n. Hence | f(x)−F_n(x) |<є for a large n independent of x.

We almost finished the demonstration that e_n(t)=(2π)^−1/2e^int is an orthonormal basis of L₂[−π,π]: