Trying to understand the differences between basic properties of $\mathbb{L}^2$ norm and the absolute value in $\mathbb{R}$ (if any exist)

Question

Suppose that $f,g:A\subset\mathbb{R}\rightarrow\mathbb{R}$ is real value function, then it holds that $$(f(x)+g(x))^2=|f(x)+g(x)|^2\leq(|f(x)|+|g(x)|)^2$$ Also, suppose that $X,Y\in\mathbb{L}^2(\Omega,\mathbb{F}_{s},\mathbb{P})$ and $||\cdot||_{\mathbb{L}^2}:V\subset\mathbb{L}^2\rightarrow\mathbb{R}^{+}$ stands for the L-square norm and $f,g:X\times Y\rightarrow \mathbb{R}$.

Is the property of the absolute value in $\mathbb{R}$ satisfied in the following way in $\mathbb{L}^2$? $$(f(X,Y)+g(X,Y))^2=||f(X,Y)+g(X,Y)||_{\mathbb{L}^2}^2\leq\left(||f(X,Y)||_{\mathbb{L}^2}+||g(X,Y)||_{\mathbb{L}^2}\right)^2$$

Answer 1

$$(f(X,Y) + g(X,Y))^2 = \| f(X,Y) + g(X,Y)\|_{L^2}^2$$ does not make sense because the left-hand side is a function whereas the right-hand side is a number.

On the other hand, the triangle inequality says $$\|f(X,Y) + g(X,Y)\|_{L^2} \leq \|f(X,Y)\|_{L^2} + \|g(X,Y)\|_{L^2},$$ which, upon squaring, becomes the desired inequality.