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Continuity of Inner product
Inner product of two continuous maps is continuous
令 $S \subseteq \mathbb{R}^n$, $C(S)$是从$S\to\mathbb{R}^m$的连续函数的集合。
令 $f,g∈C(S)$. 定义$f$与$g$的内积$h:S\times S \to \mathbb{R}$为$h(x) = \langle f(x),g(x)\rangle$. 证明: $h$是连续的。
使用连续性的序列定义(Heine definition of continuity)。也就是说，设$S$中的序列$f_n \to f, g_n \to g$，来证明$h_n =\langle f_n, g_n\rangle \to h = \langle f, g\rangle$.
\begin{aligned}
\lvert\langle f_n, g_n\rangle - \langle f, g\rangle\rvert &= \lvert\langle f_n, g_n\rangle - \langle f_n, g\rangle + \langle f_n, g\rangle - \langle f ,g\rangle\rvert\\
&\leq \lvert\langle f_n, g_n\rangle - \langle f_n, g\rangle\rvert + \lvert\langle f_n, g\rangle - \langle f,g\rangle\rvert\\
&\leq \lvert\langle f_n, g_n- g\rangle\rvert + \lvert\langle f_n- f,g\rangle\rvert\\
\small\text{根据柯西不等式，}&\leq \lVert f_n\rVert \cdot \lVert g_n-g\rVert + \lVert f_n-f\rVert \cdot \lVert g\rVert
\end{aligned}
由于收敛，$\lVert f_n\rVert$ 有界。所以$\lvert\langle f_n, g_n\rangle - \langle f, g\rangle\rvert\to0$ 当 $\lVert f_n-f\rVert $与$\lVert g_n-g\rVert \to0$. |
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