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Problem 68: V is the vector space of all real-valued functions defined on the interval $(-\infty,\infty)$, and S is the subset of V consisting of all functions satisfying $f(-x) = f(x)$ for all $(-\infty,\infty)$

Problem 68: V is the vector space of all real-valued functions defined on the interval $(-\infty,\infty)$, and S is the subset of V consisting of all functions satisfying $f(-x) = f(x)$ for all $(-\infty,\infty)$. Determine whether it is a subspace of the given vector space $\V$.

Solution:
$S= \{f \in V : f(-x) = f(x)$ for all $x \in R\}$,
where V is the vector space of all real-valued functions defined on $(-\infty,\infty)$.
If $f, g \in S$, then
$(f + g)(-x) = f(-x) + g(-x) = f(x)+g(x) = (f+g)(x)$,
which shows that $f + g \in S$.
and If $c \in R$, then
$(cf)(-x) = cf(-x) = cf(x) = (cf)x$,
which shows that $cf \in S$.
Therefore S is a subspace of V

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