Definition. Let N denote the set of positive integers, i.e., N = {1, 2, 3, ...}. A sequence is a function from the set N to the set R of real numbers. Sometimes when s is a sequence, one writes s_{n} for the value of the function s at the integer n, and at other times one writes {s_{1}, s_{2}, s_{3}, ...} to introduce s as a sequence.
The set S of all sequences forms a vector space when addition of sequences is defined by
and when the multiplication of a sequence s by a scalar c is defined by (c s)_{n} = c s_{n} , n = 1, 2, 3, ... .
Recall that a sequence s is called convergent if it has a ``limit'', i.e., if there is a number s_{*} with the property that for any given r > 0 one can find a number k(r) in N such that | s_{n} - s_{*} | < r for all n >= k(r) .
A convergent sequence s has one and only one limit s_{*}.
The set S_{c} of all convergent sequences is a linear subspace of the vector space of sequences.
Neither S nor S_{c} is finite-dimensional.
The notations S for the vector space of all sequences and S_{c} for the subspace of all convergent sequences are used in the exercises below. Let i denote the inclusion map i(s) = s from S_{c} to S.
Verify that the map lim from S_{c} to R which assigns to each convergent sequence its limit is a linear map.
Propose a definition for the term null sequence.
What are the kernel and the image of the linear map lim ?
Define a linear map (known as ``left shift'') lambda from S to S by lambda(s)_{n} = s_{n + 1} , for n = 1, 2, 3, ... .
Verify that lambda is a linear map.
What is the kernel of lambda ?
What is the image of lambda ?
Show that lim \circ lambda \circ i = lim .
Define a linear map (known as ``right shift'') rho from S to S by rho(s)_{n} = s_{n - 1} , for n = 2, 3, ... with rho(s)_{1} = 0.
Verify that rho is a linear map.
What is the kernel of rho ?
What is the image of rho ?
Show that lim \circ rho \circ i = lim .
What are the compositions lambda \circ rho and rho \circ lambda ?
Let delta_{n} be the sequence, depending on n when n is in N, that is defined by assigning the value 0 to every positive integer m with m <> n and assigning the value 1 to the integer n. That is, delta_{n} is the sequence { 0, 0, ... , 0, 1, 0, 0, ... } with the unique 1 appearing in the n^{th} slot. Show that delta_{1}, delta_{2}, ..., delta_{100} are linearly independent.
Does the infinite sum of sequences delta_{1} + delta_{2} + delta_{3} + ... make sense as a sequence? Does any infinite sum of convergent sequences make sense as a sequence?