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Problem 66: V is the vector space of all real-valued functions defined on the interval $[a, b]$, and S is the subset of V consisting of all functions satisfying $f(a) = f(b)$

Problem 66: V is the vector space of all real-valued functions defined on the interval $[a, b]$, and S is the subset of V consisting of all functions satisfying $f(a) = f(b)$. Determine whether it is a subspace of the given vector space $\V$

Solution:
$S= \{f \in V : f(a) = f(b)\}$,
where V is the vector space of all real-valued functions defined on $[a, b]$.
Note that $S \ne \varnothing$ ; since the zero function $O(x) = 0$ for all $x$ belongs to $S$.
Closure under Addition:
If $f, g \in S$, then
$(f + g)(a) = f(a) + g(a) = f(b) + g(b) = (f + g)(b)$,
which shows that $f + g \in S$.
Closure under Scalar Multiplication:
If $k \in R$ and $f \in S$, then
$(kf)(a) = kf(a) = kf(b) = (kf)(b)$,
which shows that $kf \in S$.
Therefore S is a subspace of V

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