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Definition: Finite Sum of Real-Valued Functions

$(f^{(n)})^N_{ n =1} : X \to \R$, we define the finite sum $\sum_{ i=1 }^{ N } f^{(i)} : X \to \R $ by

$$(\sum_{ i=1 }^{ N } f^{(i)}) (x) \triangleq \sum_{ i=1 }^{ N } f^{(i)}(x)$$

Definition: Infinite Series

$(X, d_X), (f^{(n)})^\infty_{ n =1} ,f : X \to \R.$

If the partial sums $\sum_{ n = 1 }^{ N} f^{(n)} $ converge pointwise to $f$ on $X$ as $N \to \infty$, we define the infinite series $\sum_{ n=1 }^{ \infty } f^{(n)} $ converges pointwise to $f $, and write $f = \sum_{ n=1 }^{ \infty } f^{(n)}$.

If the partial sums $\sum_{ n=1 }^{ N } f^{(n)}$ converge uniformly to $f$ on $X $ as $N \to \infty $, we say that the infinite series $\sum_{ n=1 }^{ N} f^{(n)} $ converge uniformly to $f $ on $X$ as $N \to \infty$, we say that the inifite series $\sum_{ n=1 }^{ \infty } f^{(n)} $ converges uniformly to $f $, and again write $f = \sum_{ n=1 }^{ \infty} f^{(n)}$.

Remarks

A series $\sum_{ n=1 }^{ \infty } f^{(n)} $ converges pointwise to $f $ on $X $ if and only if $\sum_{ n=1 }^{ \infty } f^{(n)}(x) $ converges to $f(x) $ for every $x \in X $.

Theorem

$(X, d) $, $(\R, d_{l^2}) $.

If $(f^{(k)})^\infty_{ k =1} : X \to \R$ is a collection of continuous bounded functions such that $\sum_{ k = 1 }^{\infty} \norm{f^{(k)}}_ \infty $ converges. Then there exists a continuous bounded function $f : X \to \R $ such that $\sum_{k = 1}^{\infty} f^{(k)}$ converges uniformly to $f$.