Part 2 Vector spaces of functions part:vector-spaces The main goal for this part of the course is to establish a framework where we can treat functions in the same way that we treated vectors in the multivariable course. To do this, we first introduce the idea of a vector space, a generalization of the Euclidean space Rn. While there are many interesting examples of vector spaces, we focus primarily on collections of functions. We show how many features of the Euclidean space Rn can be generalized to these other vector spaces. CHAPTER 5 Introduction to vector spaces 5.1. Vector spaces From our multivariable calculus course, we know many things we can do with vectors. The most basic properties of vectors in Euclidean space is that they can be scaled and can be added. This motivates the following definition. A collection V DefineRealVectorSpace of mathematical objects is called a real vector space if (1) for any two items v1 and v2 in V the sum v1 + v2 is also in V, (2) for any item v in V and real number ↵ the rescaled item ↵v is also in V, and if all the “usual rules” (associativity, commutativity, distributive property, etc.) hold true; see Exercise 5.1. Here are some examples. E 5.1. The collection of vectors in Rn forms a real vector space. E 5.2. The collection of all polynomials p(x) such that p(1) = 0 forms a vector space. To see this, notice that if p1 and p2 are such polynomials, then p(x) = p1(x)+p2(x) is also a polynomial such that p(1) = 0. Similarly, if p(1) = 0, then any multiple of p is also a polynomial having that property. E 5.3. The collection of all continuous functions R R forms a vector ! space that is denoted C0(R). E 5.4. The collection of all functions R R with continuous derivative ! forms a vector space that is denoted C1(R). VS:ode-solutions E 5.5. The collection of solutions u(t) to the differential equation d2u + 17u = 0 dt2 form a vector space. (In differential equations course, the fact that the space of solutions is a vector space is as the linearity principle or the principle of superpo- sition.) 33 5.1. VECTOR SPACES 34 define-l2 2 E 5.6. The collection of sequences ak such that the sum 1k=1(ak ) con- verges is a vector space, and is given the l2(N). P 2 To verify that it is a vector space, we need to verify that if ak and bk are in l (N) and ↵ is a real number, then the sums 1 2 1 2 (↵ak ) and (ak + bk ) Xk=1 Xk=1 converge. To see this, we consider the Nth partial sums, noting that N N 2 2 2 (↵ak ) ↵ (ak ) Xk=1 Xk=1 N N N 2 2 2 (ak + bk ) 2 (ak ) + 2 (bk ) . Xk=1 Xk=1 Xk=1 2 The desired convergences now follows from the fact that ak and bk are in l (N). We remark that one may also consider sequences ak where the index k rangers over all integers. The corresponding vector space is denoted l2(Z). L2-Interval-VectorSpace E 5.7. The collection of functions u:[ 1, 1] R such that − ! 1 u(x) 2dx < (5.1) L2IntervalCondition 1 | | 1 Z− is a vector space. This vector space is given the symbol L2([ 1, 1]); see Exercise 5.3. − Notice that u(x) does not need to be continuous – for example the following function is in L2([ 1, 1]): − 1/4 x− if > 0, u(x) = >80 if x 0. <> > : These examples show that the concept of a vector space is very general; indeed in linear algebra class one learns many tools for studying general vector spaces. We make these remarks: Example 5.5 is a hint that the concept of a vector space is very useful for • studying differential equations. We sometimes refer to the objects in a vector space V as “vectors”, even • when they are other types of mathematical gadgets. The numbers ↵ are sometimes called “scalars” as they serve to rescale the vectors in V. 5.2. SUBSPACES 35 We use the convention that vectors (and vector-valued functions) in Rn • are typeset in boldface. However, vectors in other vector spaces are not distinguished in any way. Pay close attention to contextual clues in order to determine which objects are being considered as vectors. Sometimes it is convenient to use complex numbers for scalars; in this • case we say that V is a complex vector space. It is also useful to consider some examples of collections of objects that are not vector spaces. E 5.8. The collection of unit vectors (i.e. vectors of magnitude 1) in Rn is not a vector space. This is straightforward to see: If two vectors each have magnitude 1, then their sum will (almost always) not have magnitude 1. Furthermore, if we rescale a unit vector, then it is no longer unit! E 5.9. The collection of polynomials p(x) such that p(1) = 2 is not a vector space. If polynomials p1(x) and p2(x) have this property, then their sum is a polynomial, but will not have the desired property. E 5.10. The collection of solutions u(t) to the differential equation d2u + 17u = cos t dt2 does not form a vector space. (Recall from differential equations class that inho- mogeneous equations satisfy a somewhat more complicated version of the super- position principle.) 5.2. Subspaces Suppose that v1,...,vk are elements of some vector space V. For any scalars ↵1,...,↵k , the quantity ↵ v + + ↵k vk 1 1 ··· is called a linear combination of the objects v1,...,vk . The collection of all possible linear combinations is called the span of the vectors, and is denoted span v ,...,vk . { 1 } 2 E 5.11. The span of the polynomials p0(x) = 1, p1(x) = x, and p2(x) = x is the collection of all polynomials of degree less than or equal to 2. 5.2. SUBSPACES 36 An important fact is that is that the span of any collection of vectors is itself a vector space. Since all of the vectors in this vector space are also in the vector space V, we say that span v ,...,vk is a “subspace” of V. { 1 } It is important that the phrase “linear combination” only refers to finite sums! Here are two examples. E 5.12. Consider the vectors v = 1, 1, 1 and v = 1, 0, 1 in the vector 1 h i 2 h − i space R3. The subspace span v , v is a plane in R3; the equation of the plane is { 1 2} x 2y + z = 0. − VS:poly-degree-three E 5.13. The collection of polynomials of degree less than 4 is span 1, x, x2, x3 , { } a subspace of the vector space of polynomials. More generally, we say that a vector space W is a subspace of vector space V if W is a vector space, and • all objects in W are also in V. • E 5.14. The collection of all functions u(x) such that u(0) = 0 is a subspace of C0(R). The following is a very important example. E 5.15. A function u:[ 1, 1] R is said to be piecewise continuous if − ! u is continuous, except at a finite number of points in [ 1, 1]. Furthermore, the − function is required to be bounded. The collection of piecewise continuous functions u:[ 1, 1] R such that the − ! integral (5.1) is finite forms a subspace of L2([ 1, 1]). We denote this subspace by − L2 ([ 1, 1]). PC − An example of a piecewise continuous function is the following: 1 if x 0, u(x) = ≥ >8 1 if x < 0. <>− > The function u(x) defined in Example: 5.7 is not an example of a piecewise contin- uous function, because it is not bounded. A type of subspace that is important in this class is a subspace defined by imposing a boundary condition. The following example appears repeatedly in this class. 5.3. FINITE AND INFINITE DIMENSIONAL VECTOR SPACES 37 ex:PeriodicTrig E 5.16 (P ). A function u:[ 1, 1] R is − ! said to satisfy periodic boundary conditions if u( 1) = u(1) and u0( 1) = u0(1). − − The reason that these conditions are called “periodic” is that any function satisfying these conditions can be periodically extended to a larger domain in such a way that both the function, and its first derivative, are continuous. Functions that satisfy periodic boundary conditions and also satisfy the condition (5.1) form a subspace of L2 ([ 1, 1]). We explore this subspace via the following PC − activities. (1) Show that the function f (x) = x3 x satisfies periodic boundary condi- − tions. Draw a sketch of the graph of f on the interval [ 1, 1]. Then extend − the graph to a larger domain by periodic repeating. Plot the periodic extension of f 0(x) on this extended domain. (2) For what values of ! does the function sin(!x) satisfy the periodic bound- ary condition? (3) For what values of ! does the function cos(!x) satisfy the periodic boundary condition? Two other boundary conditions – the Dirichlet and Neumann conditions – are explored in the exercises. 5.3. Finite and infinite dimensional vector spaces Consider the vector space of all polynomials, and let W = span 1, x, x2, x3 be the { } subspace from Example 5.13.