Linear Combination of Continuous Functions valued in Topological Vector Space is Continuous

Theorem

Let $X$ be a topological space.

Let $K$ be a topological field.

Let $Y$ be a topological vector space over $K$.

Let $n \in \N$.

Let $f_1, \ldots, f_n : X \to Y$ be continuous functions.

Let $\alpha_1, \ldots, \alpha_n \in K$.

Then:

$\ds \sum_{k \mathop = 1}^n \alpha_k f_k$ is a continuous function.

Proof

We do induction on the number of functions $n$:

Basis for the Induction

For each $\lambda \in K$, define $c_\lambda : Y \to Y$ by:

$\map {c_\lambda} y = \lambda y$

for each $y \in Y$.

From Dilation Mapping on Topological Vector Space is Continuous, $c_\lambda$ is continuous for each $\lambda \in K$.

Then we have:

$\alpha_1 f_1 = c_{\alpha_1} \circ f_1$

Since $f_1$ is continuous, $\alpha_1 f_1$ is continuous by Composite of Continuous Mappings is Continuous.

So shown for basis for the induction.

Induction Hypothesis

This is the induction hypothesis:

$\ds \sum_{k \mathop = 1}^n \alpha_k f_k$ is a continuous function.

Now we want to show that if $f_{n + 1} : X \to Y$ is another continuous function, we have:

$\ds \sum_{k \mathop = 1}^{n + 1} \alpha_k f_k$ is continuous.

Induction Step

This is the induction step.

Write:

$\ds s_n = \sum_{k \mathop = 1}^n \alpha_k f_k$

From the base case:

$\alpha_{n + 1} f_{n + 1}$ is continuous.

Define $f : X \to Y \times Y$ by:

$\map f x = \tuple {\map {s_n} x, \alpha_{n + 1} f_{n + 1} }$

for each $x \in X$.

From Continuous Mapping to Product Space, $f$ is continuous.

Define $g : Y \times Y \to Y$ by:

$\map g {x, y} = x + y$

for each $x, y \in Y$.

Since $Y$ is a topological vector space, $g$ is continuous.

From Continuous Mapping to Product Space, $g \circ f$ is continuous.

We have:

$\ds g \circ f = \sum_{k \mathop = 1}^n \alpha_k f_k + \alpha_{n + 1} f_{n + 1} = \sum_{k \mathop = 1}^{n + 1} \alpha_k f_k$

So:

$\ds \sum_{k \mathop = 1}^{n + 1} \alpha_k f_k$ is continuous.

The result follows by induction.

$\blacksquare$