Horizontal Section of Linear Combination of Functions is Linear Combination of Horizontal Sections

Theorem

Let $X$ and $Y$ be sets.

Let $f_1, f_2, \ldots, f_n : X \times Y \to \overline \R$ be functions.

Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be real numbers.

Let $y \in Y$.

Then:

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

where $f^y$ denotes the $y$-horizontal section of the function $f$.

Proof

Let $x \in X$.

We have:

\(\ds \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k}^y} x\)

\(=\)

\(\ds \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k} } {x, y}\)

Definition of Horizontal Section of Function

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \alpha_k \map {f_k} {x, y}\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \alpha_k \map {\paren {f_k}^y} x\)

Definition of Horizontal Section of Function

so:

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

$\blacksquare$