Real Analysis/Normed Linear Spaces

We will give a brief review of concepts from linear algebra regarding linear spaces and their properties. This is not an exhaustive discussion, so the reader is advised to consult a linear algebra text for more details if these topics are unfamiliar. Linear Space A linear space (also called a vector space) is a set over a field with two operations defined on , addition and scalar multiplication. Let and . The following eight properties define the structure of a linear space. {| - |1. | (Symmetry of addition) |- |2. | (Associativity of addition) |- |3. |There exists a unique element such that (Additive identity element) |- |4. |For each there exists such that (Additive inverse) |- |5. | (Scalar multiplication by multiplicative identity) |- |6. | (Associativity of scalar multiplication) |- |7. | |- |8. | |} If the field , we call this a real linear space. Similarly, if the field , we call this a complex linear space. We will restrict our study to the case of real linear spaces. Elements of a linear space (or vector space) are called vectors. Example Euclidean Space The set is the set of all n-tuples of real numbers. So for some , where , . Say and . This set is a linear space. The addition property and scalar multiplication are shown by and The reader should verify satisfies the above eight properties of a linear space. Recall, the Euclidean space is equipped with an inner product (often called the dot product in ) that is given by . This will come in handy in a following example. Subspace A subspace of a vector space is a nonempty subset such that is also a linear space. So for any and we have that (i.e., is closed under addition and scalar multiplication). Example Consider and . Then the span of is the set . We will show that the set is a subspace of . Proof First, we need to show that the zero element is in (otherwise, it could not be a linear space). It follows that . Therefore, . Now, suppose and . We see . Since by the field properties of the reals, we have that . Hence, this set is closed under the vector space operations. Therefore, is a subspace of . The astute reader should notice that this subspace is a plane in three-dimensional space. Basis A linear combination of vectors is an expression , where and for . This can be expressed in more concise notation as . Given a nonempty subset of a linear space the set of all linear combinations of elements of is called the span of , which we denote with . The span of will generate a subspace of . A collection of vectors from is said to be linearly independent when only when . If any nonzero satisfies this equation, then the set of vectors is called linearly dependent. A basis of a linear space is a linearly independent set of vectors which spans . That is, the subset is a basis if and is a linearly independent set of vectors. The number of linearly independent vectors it takes to span a vector space defines the dimension of that vector space. A vector space is finite-dimensional if it can be spanned by a finite set of basis vectors. If a vector space is not finite-dimensional, it is infinite-dimensional. We denote the dimension of a vector space as . Example The standard basis of the vector space is the set , where the -th basis vector has a one in the -th entry and zeroes elsewhere. So, in three-dimensional space our basis is the set . Hence, any vector in can be expressed as a linear combination of these basis vectors. A suggested exercise for the reader is to prove that the expression of a vector as a linear combination of basis vectors is unique. Note that the basis set has 3 elements, so the space it spans has a dimension of 3, i.e, . Norms In a linear space, we often want to have a concept of the "size" of elements or of the distance between elements. A norm is a function such that for and the following properties hold. (Triangle inequality) The norm serves as a way to describe the size of individual elements. Now, if the norm is used to describe the size of a difference between vectors (), it measures the distance between the two. So, we find that the norm induces a metric on the space . Hence, we describe a metric function on by . The reader is encouraged to verify that this function satisfies the metric properties. A linear space with a norm on it is called a normed linear space. A normed linear space that is complete is called a Banach space. Example Euclidean space has a norm given by . This should be familiar from the Pythagorean theorem. Since is complete, we have that is also complete. Thus, Euclidean space would also be an example of a Banach space. We should also note that the norm here can be expressed as . So, in this case, the inner product induces a norm on our space.