Find the function $g∈U$that minimises $\|f−g\|$

Question

Let $V=C\big([−1,1] ,\mathbb R\big)$ be the vector space of continuous real-valued functions defined on $[−1,1]$ with the usual addition and scalar multiplication of functions. Equip with the inner product

$$\langle f,g\rangle =\int_{-1}^1\ f(x)\ g(x)\ dx\,.$$

Let $f\in C\big([−1,1] ,\mathbb R\big)$ be given by $$f(x) =\begin{cases}0 &\text{if }−1\leq x\leq 0\,,\\ x &\text{if }0\leq x\leq 1\,.\end{cases}$$ Let $B = \left\{\frac1{\sqrt2}, \cos(\pi x), \sin(\pi x)\right\} \subseteq V$ and $U = \text{span}(B)$.

Find the function $g∈U$that minimises $\|f−g\|$, i.e., $$\|f−g\|≤\|f−h\| \text{ for all }h\in U\,.$$ Not sure of this one im a bit confused and no where near the answer

Answer 1

$B$ is an orthonormal set. The answer is $\frac a {\sqrt 2} +b\cos (\pi x)+c \sin (\pi x)$ where $a=\langle f, \frac 1 {\sqrt 2} \rangle, b=\langle f, \cos (\pi x) \rangle, c=\langle f, \sin (\pi x) \rangle$. For example, $a=\int_0^{1} x \frac 1 {\sqrt 2} dx=\frac 1 {2\sqrt 2}$.