Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

This chapter develops the beginnings of abstract functional analysis, a subject designed to study properties of functions by treating the functions as the members of a space and formulating the properties as properties of the space. Section 1 defines Banach spaces as complete normed linear spaces and gives a number of examples of these. The space of bounded linear operators from one normed linear space to another is a normed linear space, and it is a Banach space if the range is a Banach space. Sections 2–3 concern Hilbert spaces. These are Banach spaces whose norms are induced by inner products. Section 2 shows that closed vector subspaces of such a space have orthogonal complements, and it shows the role of orthonormal bases for such a space. Section 3 concentrates on bounded linear operators from a Hilbert space to itself and constructs the adjoint of each such operator. Sections 4–6 prove the three main abstract theorems about the norm topology of general normed linear spaces—the Hahn–Banach Theorem, the Uniform Boundedness or Banach–Steinhaus Theorem, and the Interior Mapping Principle. A number of consequences of these theorems are given. The second and third of the theorems require some hypothesis of completeness. The topic of Hilbert and Banach spaces continues in Chapter IV of the companion volume, Advanced Real Analysis.

The application of mathematics to natural phenomena often brings up the question : when a linear equation is changed slightly, how much of a change results in the solution? Here vector is used to include, for example, vectors in Hubert space and elements of (linear) function spaces; also it is assumedC) that the equation is uniquely solvable. We formulate and answer the question in §2 for a linear equation in a Banach space, then specialize to deduce perturbation limits for linear algebraic systems, infinite systems, and integral equations. Finally we obtain error limitations for certain approximate methods of solving infinite linear systems (method of segments) and integral equations (method of Goursat-Schmidt). It is hoped that these limitations may be useful to applied mathematicians. The methods and results of this paper unify and extend investigations concerning algebraic systems by F. R. Moulton [15](2), Etherington [6] and the author [13, 14]; and on integral equations of Fredholm type and second kind by Tricomi [21]. However, they do not cover perturbation questions associated with characteristic values and characteristic functions, which have been studied by Lord Rayleigh [16, p. 115], Courant [4, p. 296] and Mrs. Adams [l]. §1 is expository, containing as much about abstract spaces as is needed for §2. 1. Vector spaces. It will be useful to collect here some facts about normed linear spaces ispaces L), Banach spaces ispaces B), and linear transformations. A space L is linear: if xEL and a is any complex number, the product ax is defined and axEL; if also y EL, the sum x+y is defined and x+yEL. With each element x of L is associated a non-negative real number ||*||, its norm; ||*|| >0 unless x = @, the zero-element of L; |J@|| =0. The norm has properties of an absolute value: ||a*|| = \a -11*11, ||*+y|| a||*||+||y||(We have described a complex space L; in a real space, number a must be real.) A normed linear space is a Banach space 5 [2, p. 53] if it is furthermore complete: if {*„} is an infinite sequence of elements of 5, and if II*»—*»||—*0 as m, m—»oo, there exists a xo of 5 such that ||*„—*o||—»0 as n—+ oo (strong completeness). Examples of such spaces are listed in Banach's book [2, pp. 10-12, examples 3-10], and several occur in the remainder of this paper.

There are many characterizations which distinguish finite-dimensional normed linear spaces from infinite-dimensional normed linear spaces. Perhaps the best known of these is the compactness of the unit ball. Recently, V. Klee [1] showed that in any infinite-dimensional normed linear space there exists a decreasing sequence of unbounded but linearly bounded closed convex sets whose intersection is empty. We will give here a somewhat similar condition which holds in all infinite-dimensional normed linear spaces but does not hold in any finite-dimensional space. We begin with the following terminology. U will denote the unit ball {x: ||x|| S 1 } and S the unit sphere {x: ||x|| = 1 }. A shell will be any set of the form {x: ri? IIx-xalI ? r2 for 0 1 such that (x+p U)Cr (-x+p U) CO, and S will be called finitely nonfiat if there exists a finite number of points {xi} '1 in S such that n(xj+pU) Co. A collection of sets has the finite intersection property if the intersection of the sets in any finite subcollection is not empty.

The combination of algebraic structures and graphs was carried out by investigating thoroughly the relation among the algebraic structure and the graph theoretic properties. Moreover, it needs to explore algebraic structure. The results of the combination of algebraic structures and graphs have many applications in the fields of Internet modeling, coding, etc. For example, the famous Cayley graph was constructed from groups and widely used in network models. Das introduced the subspace inclusion graph on finite-dimensional vector space over a finite filed, and studied that the graph is bipartite and some special properties if the dimension of the vector space is 3. In this paper, this bipartite inclusion graph in the case of 3-dimensional is extended to more general dimensional bipartite inclusion graph. The diameter, girth, clique number, covering number, independence number and matching number are studied and the properties are shown, such as regular, planar and Eulerian. Moreover, the authors also introduce a new results about the structure and the number of 1-dimensional and n-1-dimensional subspaces on n-dimensional vector space.

This chapter, in fact, starts our study of functional analysis. Here, we consider linear normed spaces, i.e., sets equipped with the structure of a linear space and a norm (an analog of the notion of the length of a vector). These spaces are one of the main objects of investigation in functional analysis. Banach spaces or spaces of type B (complete with respect to convergence in norm) are the most important class of linear normed spaces. These spaces were introduced by Banach and Wiener. Here, we also consider another important class of spaces — Hilbert spaces — whose norm is defined in terms of a scalar product.

2. Additions to IV.2. We show below that if S and S) are infinite-dimensional NLS's and if S is a conjugate space (in particular, if S is reflexive), then 53(X, 9)l is not B-convex. This implies that if the conjecture that every B-convex space is reflexive is true, then also the conjecture that every 53(f, J)) is not B-convex for infinite-dimensional S and S) is true (since, by IV.2, if S is not B-convex, then Z ,)) is not B-convex). The principal tool here is the theorem of Aryeh Dvoretzky (see, e.g., Some results on convex bodies and Banach spaces, Proceedings of the International Symposium on Linear Spaces, pp. 123-160, Pergamon Press, New York, 1961) which states that every infinite-dimensional Banach space contains arbitrarily good approximations of finite-dimensional Hilbert space of every finite dimension. Let an(.) be the sequence of period 2n which starts with 2n-I (+ 1)'s and then 2n' (-1)'s. Let k be fixed >2 and let m=2k. Denote the usual basis of 12 by {8n}. Fix E >O and use Dvoretzky's theorem to find xl, ..., xm of unit norm in S such that

Extract Although examples drive this book, we first provide a whirlwind survey of the general concepts of operator theory. We do not expect the student to master these topics now since they are covered in future chapters. A Hilbert space H is a complex vector space endowed with an inner product ⟨x,y⟩ that defines a norm ‖x‖=⟨x,x⟩ with respect to which H is (Cauchy) complete. The inner product on a Hilbert space satisfies the Cauchy–Schwarz inequality |⟨x,y⟩|⩽‖x‖‖y‖ for all x,y∈H⁠. Examples of Hilbert spaces include Cn (complex Euclidean space), ℓ2 (the space of square-summable complex sequences), and L2[0,1] (the Lebesgue space of square-integrable, complex-valued functions on [0,1]⁠). Vectors x,y in a Hilbert space H are orthogonal if ⟨x,y⟩=0⁠. The dimension of a Hilbert space H is the cardinality of a maximal set of nonzero orthogonal vectors. This book is almost exclusively concerned with Hilbert spaces of countable dimension. Every such Hilbert space has an orthonormal basis (un)n=1∞⁠, a (possibly finite) maximal orthogonal set such that ⟨um,un⟩=δmn for all m,n⩾1⁠. With respect to an orthonormal basis (un)n=1∞⁠, each x∈H enjoys a generalized Fourier expansion x=∑n=1∞⟨x,un⟩un that satisfies Parseval’s identity ‖x‖2=∑n=1∞|⟨x,un⟩|2. A subspace (a norm-closed linear submanifold) of H is itself a Hilbert space with the operations inherited from H⁠. If (wn)n=1∞ is a (possibly finite) orthonormal basis for a subspace M of H and x∈H⁠, then PMx=∑n=1∞⟨x,wn⟩wn belongs to M and satisfies ‖x−PMx‖⩽‖x−y‖ for every y∈M⁠. In short, PMx is the unique closest vector to x in M⁠. Furthermore, PM defines a linear transformation on H whose range is M⁠. It is called the orthogonal projection of H onto M and it satisfies PM2=PM and ⟨PMx,y⟩=⟨x,PMy⟩ for all x,y∈H⁠. Let H and K be Hilbert spaces. A linear transformation A:H→K is bounded if ‖A‖=sup{‖Ax‖K:‖x‖H=1} is finite. Let B(H,K) denote the set of bounded linear operators from H to K⁠. We write B(H) for B(H,H)⁠. The quantity ‖A‖ is the norm of A. Since B(H) is closed under addition and scalar multiplication, it is a vector space. Furthermore, since ‖A+B‖⩽‖A‖+‖B‖ and ‖cA‖=|c|‖A‖ for all A,B∈B(H) and c∈C⁠, it follows that B(H) is a normed vector space. Endowed with this norm, B(H) is complete and thus forms a Banach space. Moreover, the composition AB belongs to B(H) and ‖AB‖⩽‖A‖‖B‖ for all A,B∈B(H)⁠. Therefore, B(H) is a Banach algebra. For each A∈B(H,K)⁠, there is a unique A*∈B(K,H) such that ⟨Ax,y⟩K=⟨x,A*y⟩H for all x∈H and y∈K⁠. The operator A* is the adjoint of A; it is the analogue of the conjugate transpose of a matrix. One can show that A↦A* is conjugate linear, that A**=A⁠, ‖A‖=‖A*‖⁠, and ‖A*A‖=‖A‖2⁠. This additional structure upgrades B(H) from a Banach algebra to a C*-algebra. One can exploit adjoints to obtain information about the kernel and range of an operator. For most of the operators A∈B(H) covered in this book, we give the matrix representation [A]=[⟨Auj,ui⟩]i,j=1∞ with respect to an orthonormal basis (un)n=1∞ for H⁠. This matrix representation [A] defines a bounded operator x↦[A]x on the Hilbert space ℓ2 of square summable sequences that is structurally identical to A. Schur’s theorem helps us determine which infinite matrices define bounded operators on ℓ2⁠. Many of the operators covered in this book, such as the Cesàro operator, the Volterra operator, weighted shifts, Toeplitz operators, and Hankel operators, have fascinating structured-matrix representations. An important class of operators is the compact operators. These are the A∈B(H) such that (Axn)n=1∞ has a convergent subsequence whenever (xn)n=1∞ is a bounded sequence in H⁠. Equivalently, an operator is compact if it takes each bounded set to one whose closure is compact. Each finite-rank operator is compact and every compact operator is the norm limit of finite-rank operators. The compact operators form a norm-closed, *-closed ideal within B(H)⁠. Some operators have a particularly close relationship with their adjoint. For example, the operator M on L2[0,1] defined by (Mf)(x)=xf(x) satisfies M*=M⁠. Such operators are selfadjoint. If µ is a positive finite compactly supported Borel measure on C⁠, then the operator N on L2(μ) defined by (Nf)(z)=zf(z) satisfies (N*f)(z)=z―f(z)⁠, and thus N*N=NN*⁠. Such operators are normal. The operator (Uf)(eiθ)=eiθf(eiθ) on L2(T) satisfies U*U=UU*=I⁠. Such operators are unitary. Unitary operators preserve the ambient structure of Hilbert spaces and can serve as a vehicle to relate A∈B(H) with B∈B(K)⁠. We say that A is unitarily equivalent to B if there is a unitary U∈B(H,K) such that UAU*=B⁠. Unitary equivalence is often used to identify seemingly complicated operators with relatively simple ones.

Here we want to present briefly some results, problems and directions of research in the modern theory of best approximation, i.e. in which the methods of functional analysis are applied in a consequent manner. In this theory the functions to be approximated and the approximating functions are regarded as elements of certain normed linear (or, more generally, of certain metric) spaces of functions and best approximation amounts to finding “nearest points”. The advantages and a brief history of this modern point of view have been described in the Introduction to the monograph [82] and we shall not repeat them here; the material which will be presented in the sequel will be convincing enough, we hope, to prove again that the theory of best approximation in normed linear spaces constitutes both a rigorous theoretical foundation for the existing classical and more recent results in various concrete spaces and a powerfull tool for obtaining new results, solving the new problems which appear.

We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in normed linear spaces. In this context, the key aspect of our study is to consider norming sequences for a bounded linear operator, instead of norm attaining elements. We also obtain a complete characterization of smoothness of bounded linear operators between real normed linear spaces, when the corresponding norm attainment set non-empty. This illustrates the importance of the norm attainment set in the study of smoothness of bounded linear operators. Finally, we prove that Gâteaux differentiability and Fréchet differentiability are equivalent for compact operators in the space of bounded linear operators between a reflexive Kadets-Klee Banach space and a Fréchet differentiable normed linear space, endowed with the usual operator norm.

Now we reach the centerpiece of this volume, which is the theory of bases in Banach spaces. Since every Banach space is a vector space, it has a basis in the ordinary vector space sense, i.e., a set that spans and is linearly independent.However, this definition of basis restricts us to using only finite linear combinations of vectors, while in any normed space it makes sense to deal with infinite series. Restricting to finite linear combinations when working in an infinite-dimensional space is simply too restrictive for most purposes. Moreover, the proof that a vector space basis exists is nonconstructive in general, as it relies on the Axiom of Choice. Hence we need a new notion of basis that is appropriate for infinite-dimensional Banach spaces, and that is the main topic of this chapter.

Unless otherwise specified, E will be a real or complex normed linear space. A complete normed linear space is called a Banach space. E’ is the dual space, i.e., the Banach space of continuous linear functionals, equipped with the norm \(\left\| l \right\| = \mathop {\sup }\limits_{\left\| x \right\| = 1} \left| {l\left( x \right)} \right|\).KeywordsNormed Linear SpaceSmooth Banach SpaceStrong UniquenessCharacterization TheoremGaussian Quadrature FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

In this paper, we prove the generalized Hyers-Ulam stability of the additive functional inequality <TEX>$${\parallel}f(2x_1)+f(2x_2)+{\cdots}+f(2x_n){\parallel}{\leq}{\parallel}tf(x_1+x_2+{\cdots}+x_n){\parallel}$$</TEX> in Banach spaces where a positive integer <TEX>$n{\geq}3$</TEX> and a real number t such that 2<TEX>${\leq}$</TEX>t<n.

Let us remind that by a function we mean a mapping having the domain in the scalar field \( \mathbb{K} \) of real or complex numbers (see Chapter 0, §1). In this chapter, we will consider functions with values in normed spaces — called vector-functions —, as well as functions with values in the field of scalars, which are called scalar-functions. The symbol f(t) means either the function f itself, or the value of f at t; the precise meaning will follow from the context. Prom now on the term “vector-function” will be used in a broader sense, for any mapping \( f:\mathfrak{X} \to \mathfrak{Y} \) with the domain in a finite dimensional space and with values in a normed space.

Let Gp(x,y,z)=∥x+y+z∥p+∥z∥p−∥x+z∥p−∥y+z∥p be defined on a normed space X. The special case G2(x,y,z)=0,∀z∈X, where X is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where X is a complex inner product space endowed with the inner product ⟨·,·⟩ and induced norm ∥·∥, it is proved that Sgn(G2(x,y,z))=Sgn(Re⟨x,y⟩),∀z∈X, and a geometric explanation for condition Re⟨x,y⟩=0 is provided. Furthermore, a condition G2(x,iy,z)=0,∀z∈X is added to extend the T-orthogonality to the general complex normed linear spaces. Based on some characterizations, the T-orthogonality is compared with several other well-known types of orthogonality. The fact that T-orthogonality implies Roberts orthogonality is also revealed.