1  A norm on a vector space, ||x||, satisfies ||x||≥ 0, ||x||=0 if and only if x=0, ||λ x||=|λ|  ||x||, and ||x+y|| ≤ ||x|| + ||y|| (triangle inequality). A norm defines a metric and a complete normed space is called a Banach space.

 2  An inner-product space is a vector space (usually complex) with a scalar product on it, ⟨ x,y⟩ ∈ ℂ such that ⟨ x,y⟩=⟨ y,x⟩, ⟨ λ x,y⟩=λ⟨ x,y⟩, ⟨ x+y,z⟩ =⟨ x,z⟩ +⟨ y,z⟩ , ⟨ x,x⟩ ≥ 0 and ⟨ x,x⟩ =0 if and only if x=0. This defines a norm by ||x||²=⟨ x,x⟩ . A complete inner-product space is called a Hilbert space. A Hilbert space is automatically a Banach space.

 3  The Cauchy–Schwarz inequality. |⟨ x,y⟩ | ≤ ||x|| ||y|| with equality if and only if x and y are linearly dependent.

 4  Some examples of Hilbert spaces. (i) Euclidean ℂⁿ. (ii) l₂, sequences (a_k) with ||(a_k)||₂²=∑|a_k|² < ∞. In both cases ⟨ (a_k),(b_k)⟩=∑a_kb_k. (iii) L₂[a,b], functions on [a,b] with ||f||₂²=∫_a^b |f(t)|²   dt < ∞. Here ⟨ f,g ⟩=∫_a^b f(t) g(t) d t. (iv) Any closed subspace of a Hilbert space.

 5  Other examples of Banach spaces. (i) C_b(X), continuous bounded functions on a topological space X. (ii) l_∞(X), all bounded functions on a set X. The supremum norms on C_b(X) and l_∞(X) make them into Banach spaces. (iii) Any closed subspace of a Banach space.

 6  On incomplete spaces. The inner-product (L₂) norm on C[0,1] is incomplete. c₀₀ (sequences eventually zero), with the l₂ norm, is another incomplete i.p.s.

 7  The parallelogram identity. ||x+y||² + ||x−y||² = 2||x||² + 2||y||² in an inner-product space. Not in general normed spaces.

 8  On subspaces. Complete =⇒ closed. The closure of a linear subspace is still a linear subspace. Lin (A) is the smallest subspace containing A and CLin (A) is its closure, the smallest closed subspace containing A.

 9  From now on we work in inner-product spaces.

 10  The orthogonality. x ⊥ y if ⟨ x,y⟩ =0. An orthogonal sequence has ⟨ e_n,e_m⟩ =0 for n ≠ m. If all the vectors have norm 1 it is an orthonormal sequence (o.n.s.), e.g. e_n=(0,…,0,1,0,0,…) ∈ l₂ and e_n(t)=(1/√2π) e^int in L₂(−π,π).

 11 Pythagoras’s theorem: if x⊥ y then ||x+y||²=||x||²+||y||².

 12  The best approximation to x by a linear combination ∑_k=1ⁿλ_ke_k is ∑_k=1ⁿ ⟨ x,e_k⟩ e_k if the e_k are orthonormal. Note that ⟨ x,e_k⟩ is the Fourier coefficient of x w.r.t. e_k.

 13  Bessel’s inequality. ||x||² ≥ ∑_k=1ⁿ |⟨ x,e_k⟩ |² if e₁,…,e_n is an o.n.s.

 14  Riesz–Fischer theorem. For an o.n.s. (e_n) in a Hilbert space, ∑λ_n e_n converges if and only if ∑|λ_n|² < ∞; then ||∑λ_n e_n ||² = ∑|λ_n|².

 15  A complete o.n.s. or orthonormal basis (o.n.b.) is an o.n.s. ( e_n) such that if ⟨ y,e_n⟩ =0 for all n then y=0. In that case every vector is of the form ∑λ_n e_n as in the R-F theorem. Equivalently: the closed linear span of the (e_n) is the whole space.

 16  Gram–Schmidt orthonormalization process. Start with x₁, x₂, … linearly independent. Construct e₁, e₂, … an o.n.s. by inductively setting y_n+1=x_n+1−∑_k=1ⁿ ⟨ x_n+1,e_k⟩ e_k and then normalizing e_n+1=y_n+1/||y_n+1||.

 17  On orthogonal complements. M^⊥ is the set of all vectors orthogonal to everything in M. If M is a closed linear subspace of a Hilbert space H then H=M ⊕ M^⊥. There is also a linear map, P_M the projection from H onto M with kernel M^⊥.

 18  Fourier series. Work in L₂(−π,π) with o.n.s. e_n(t)=(1/√2π)e^int. Let CP(−π,π) be the continuous periodic functions, which are dense in L₂. For f ∈ CP(−π,π) write f_m=∑_n=−m^m ⟨ f,e_n⟩ e_n, m ≥ 0. We wish to show that ||f_m−f||₂ → 0, i.e., that (e_n) is an o.n.b.

 19  The Fejér kernel. For f∈ CP(−π,π) write F_m=(f₀+…+f_m)/(m+1). Then F_m(x)=(1/2π) ∫_−π^πf(t) K_m(x−t)  d t where K_m(t)=(1/(m+1)) ∑_k=0^m ∑_n=−k^k e^int is the Fejér kernel. Also K_m(t)=(1/(m+1)) [sin² (m+1)t/2] / [sin² t/2].

 20  Fejér’s theorem. If f ∈ CP(−π,π) then its Fejér sums tend uniformly to f on [−π,π] and hence in L₂ norm also. Hence CLin ((e_n)) ⊇ CP(−π,π) so must be all of L₂(−π,π). Thus (e_n) is an o.n.b.

 21  Corollary. If f ∈ L₂(−π,π) then f(t)=∑c_n e^int with convergence in L₂, where c_n=(1/2π) ∫_−π^πf(t)e^−int d t.

 22  Parseval’s formula. If f, g∈ L₂(−π,π) have Fourier series ∑c_n e^int and ∑ d_n e^int then (1/2π)⟨ f,g⟩ = ∑c_n d_n.

 23  Weierstrass approximation theorem. The polynomials are dense in C[a,b] for any a<b (in the supremum norm).

 24  On dual spaces. A linear functional on a vector space X is a linear mapping α:X → ℂ (or to ℝ in the real case), i.e., α(ax+by)=aα(x)+bα(y). When X is a normed space, α is continuous if and only if it is bounded, i.e., sup{|α(x)|: ||x|| ≤ 1} < ∞. Then we define ||α|| to be this sup, and it is a norm on the space X^* of bounded linear functionals, making X^* into a Banach space.

 25  Riesz–Fréchet theorem. If α:H → ℂ is a bounded linear functional on a Hilbert space H, then there is a unique y ∈ H such that α(x)=⟨ x,y⟩ for all x ∈ H; also ||α||=||y||.

 26  On linear operator. These are linear mappings T: X → Y, between normed spaces. Defining ||T||=sup{||T(x)||: ||x|| ≤ 1}, finite, makes the bounded (i.e., continuous) operators into a normed space, B(X,Y). When Y is complete, so is B(X,Y). We get ||Tx|| ≤ ||T||   ||x||, and, when we can compose operators, ||ST|| ≤ ||S||   ||T||. Write B(X) for B(X,X), and for T ∈ B(X), ||Tⁿ|| ≤ ||T||ⁿ. Inverse S=T⁻¹ when ST=TS=I.

 27  On adjoints. T ∈ B(H,K) determines T^* ∈ B(K,H) such that ⟨ Th, k ⟩_K = ⟨ h, T^*k ⟩_H for all h ∈ H, k ∈ K. Also ||T^*||=||T|| and T^**=T.

 28  On unitary operator. Those U ∈ B(H) for which UU^*=U^*U=I. Equivalently, U is surjective and an isometry (and hence preserves the inner product).

Hermitian operator or self-adjoint operator. Those T ∈ B(H) such that T=T^*.

On normal operator. Those T ∈ B(H) such that TT^*=T^*T (so including Hermitian and unitary operators).

 29  On spectrum. σ(T)={λ ∈ ℂ: (T−λ I) is not invertible in B(X)}. Includes all eigenvalues λ where Tx=λ x for some x ≠ 0, and often other things as well. On spectral radius: r(T)=sup{|λ|: λ∈ σ(T)}. Properties: σ(T) is closed, bounded and nonempty. Proof: based on the fact that (I−A) is invertible for ||A|| < 1. This implies that r(T) ≤ ||T||.

 30  The spectral radius formula. r(T)=inf_{n ≥ 1} ||Tⁿ||^1/n = lim_{n → ∞} ||Tⁿ||^1/n.

Note that σ(Tⁿ)={λⁿ: λ ∈ σ(T)} and σ(T^*)={λ: λ ∈ σ(T)}. The spectrum of a unitary operator is contained in {|z|=1}, and the spectrum of a self-adjoint operator is real (proof by Cayley transform: U=(T−iI)(T+iI)⁻¹ is unitary).

 31  On finite rank operator. T∈ F(X,Y) if Im  T is finite-dimensional.

On compact operator. T ∈ K(X,Y) if: whenever (x_n) is bounded, then (Tx_n) has a convergent subsequence. Now F(X,Y) ⊆ K(X,Y) since bounded sequences in a finite-dimensional space have convergent subsequences (because when Z is f.d., Z is isomorphic to l₂ⁿ, i.e., ∃ S:l₂ⁿ → Z with S, S⁻¹ bounded). Also limits of compact operators are compact, which shows that a diagonal operator Tx=∑λ_n⟨ x,e_n ⟩ e_n is compact iff λ_n → 0.

 32  Hilbert–Schmidt operators. T is H–S when ∑ ||Te_n||² < ∞ for some o.n.b. (e_n). All such operators are compact—write them as a limit of finite rank operators T_k with T_k∑_n=1^∞a_ne_n=∑_n=1^k a_n (Te_n). This class includes integral operators T: L₂(a,b)→ L₂(a,b) of the form

(Tf)(x)=

b ∫ a

K(x,y) f(y) d y,

where K is continuous on [a,b] × [a,b].

 33  On spectral properties of normal operators. If T is normal, then (i) kerT=kerT^*, so Tx=λ x =⇒ T^*x=λx; (ii) eigenvectors corresponding to distinct eigenvalues are orthogonal; (iii) ||T||=r(T).

If T ∈ B(H) is compact normal, then its set of eigenvalues is either finite or a sequence tending to zero. The eigenspaces are finite-dimensional, except possibly for λ=0. All nonzero points of the spectrum are eigenvalues.

 34  On spectral theorem for compact normal operators. There is an orthonormal sequence (e_k) of eigenvectors of T, and eigenvalues (λ_k), such that Tx=∑_k λ_k ⟨ x,e_k ⟩ e_k. If (λ_k) is an infinite sequence, then it tends to 0. All operators of the above form are compact and normal.

Corollary. In the spectral theorem we can have the same formula with an orthonormal basis, adding in vectors from kerT.

 35  On general compact operators. We can write Tx=∑µ_k ⟨ x, e_k ⟩ f_k, where (e_k) and (f_k) are orthonormal sequences and (µ_k) is either a finite sequence or an infinite sequence tending to 0. Hence T ∈ B(H) is compact if and only if it is the norm limit of a sequence of finite-rank operators.

 36  On integral equations. Fredholm equations on L₂(a,b) are Tφ=f or φ−λ Tφ=f, where (Tφ)(x)=∫_a^b K(x,y)φ(y) d y. Volterra equations similar, except that T is now defined by (Tφ)(x)=∫_a^x K(x,y)φ(y) d y.

 37  Neumann series. (I−λ T)⁻¹=1+λ T+λ² T² + …, for ||λ T ||<1.

On separable kernel. K(x,y)=∑_j=1ⁿ g_j(x)h_j(y). The image of T (and hence its eigenvectors for λ≠ 0) lies in the space spanned by g₁,…,g_n.

 38  Hilbert–Schmidt theory. Suppose that K ∈ C([a,b]× [a,b]) and K(y,x)=K(x,y). Then (in the Fredholm case) T is a self-adjoint Hilbert–Schmidt operator and eigenvectors corresponding to nonzero eigenvalues are continuous functions. If λ≠ 0 and 1/λ ∉σ(T), the the solution of φ−λ Tφ=f is

φ=

∞ ∑ k=1

⟨ f,vk ⟩ 1−λλk

vk.

 39  Fredholm alternative. Let T be compact and normal and λ≠ 0. Consider the equations (i) φ−λ Tφ=0 and (ii) φ−λ Tφ=f. Then EITHER (A) The only solution of (i) is φ=0 and (ii) has a unique solution for all f OR (B) (i) has nonzero solutions φ and (ii) can be solved if and only if f is orthogonal to every solution of (i).