In Exercises 1 through 28, determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The functions in Exercises 1 through 17 are real-valued. In Exercises 3, 4, and 5, each function has domain containing 0 and 1. In Exercises 7 through 12, each domain contains all real numbers.
20. All convergent real series.
Recall from Volume 1 Section 10.5 the definition of an infinite series:
From a given sequence of realor complex numbers, we can always generate a new sequence by adding together successive terms. Thus, if the given sequence has the terms
$$
a_1,\ a_2,\ ..., a_n,\ ...
$$we may form, in succession, the "partial sums"
$$
s_1 = a_1,
\quad
s_2 = a_1 + a_2,
\quad
s_3 = a_1 + a_2 + a_3,
$$and so on, the partial sum $s_n$ of the first $n$ terms being defined as follows:
\begin{align*}
s_n = a_1 + a_2 + \cdots + a_n &= \sum_{k = 1}^n a_k.
\end{align*}The sequence $\{s_n\}$ of partial sums is called and infinite series, or simple a $series.$ If there is a real or complex number $S$ such that
$$
\lim_{n \to \infty}s_n = S
$$we say that the series $\sum_{k = 1}^{\infty}a_k$ is convergent and has the sum $S,$ in which case we write:
$$
\sum_{k = 1}^{\infty}a_k = S.
$$
Suppose we have two real-valued series whose respective terms are $a_k$ and $b_k$ for $k = 1, 2, 3, ...$ and whose respective $n^{th}$ partial sums are $s_n$ and $t_n$. If they are convergent, then for some real $S$ and $T,$ we have
\begin{align*}
\\
\lim_{n \to \infty}s_n &= S,
\quad
\text{and}
\quad
\lim_{n \to \infty}t_n = T,
\end{align*}
Now, suppose we have another series whose $k^{th}$ term is $c_k = a_k + b_k$ and whose $n^{th}$ partial sum is $g_n = s_n + t_n.$ But as a consequence of Theorem 3.1 (i) [Volume 1, Section 3.3], $g_n \to S + T$ as $n \to \infty,$ which means that $\sum_{k = 1}^{\infty}c_k$ is convergent. Moreover, this means the set of convergent real series satisfies closure under addition.
To verify closure under multiplication by real numbers, we refer to Theorem 3.1 (iii). If $a$ is some real constant, and $\sum_{k = 1}^{\infty}s_k = S,$ then we have:
\begin{align*}
\sum_{k = 1}^{\infty}a\cdot s_k &= a\sum_{k = 1}^{\infty}s_k
\\
&= aS
\end{align*}Then, if $a = -1,$ we can verify that $\sum_{k = 1}^{\infty}-s_k$ is also convergent, with
\begin{align*}
\sum_{k = 1}^{\infty}s_k - s_k &= 0
\end{align*}thus verifying the existence of negatives in the set. The zero element is the series whose elements are $z_k = 0$ for all $k$ and whose sum is trivially $0.$
Now, since every convergent series of real-valued terms has a real-valued sum, we can treat each sum (and partial sum) as a real number when verifying axioms regarding addition and multiplication by real numbers. As such, we have shown that the set of all convergent real-valued series is a real linear space. $\,\blacksquare$
