1. Projection onto a line in $\mathbb{R}^2$
Choose a vector $y$ and a line direction $u$. The projection is $$\operatorname{Proj}_{\operatorname{span}(u)}y=\frac{\langle y,u\rangle}{\langle u,u\rangle}u.$$
Vector $y=(y_1,y_2)$
Direction $u=(u_1,u_2)$
2. Geometry of projection
Blue: $y$. Orange: projection $p$. Dashed: residual $y-p$, orthogonal to the line.
3. Best polynomial approximation to $e^x$
Approximate $e^x$ on $[0,1]$ in the $L^2$ inner product $$\langle f,g\rangle=\int_0^1 f(x)g(x)\,dx.$$
4. Fourier partial sums
The square wave has Hilbert-space coordinate expansion $$f(x)\sim \frac4\pi\left(\sin x+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}+\cdots\right).$$
5. $\ell^2$ sequence test
A sequence $x=(x_n)$ is in $\ell^2$ if $$\sum_{n=1}^{\infty}|x_n|^2<\infty.$$
6. Kernel Gram matrix
For points $x_1,x_2,x_3$, the polynomial kernel $$K(x,y)=(1+xy)^2$$ produces a Gram matrix $G=[K(x_i,x_j)]$.
7. Independent-study tasks with answers
Answer. $\langle y,u\rangle=5$, $\langle u,u\rangle=2$, so $p=\frac52(1,1)=(2.5,2.5)$.
Answer. By construction, $p=\frac{\langle y,u\rangle}{\langle u,u\rangle}u$, so $\langle y-p,u\rangle=\langle y,u\rangle-\frac{\langle y,u\rangle}{\langle u,u\rangle}\langle u,u\rangle=0$.
Answer. $1/\sqrt n$ is not in $\ell^2$, because its square is $1/n$, and the harmonic series diverges.
Answer. It is the orthogonal projection of a function onto the finite-dimensional subspace spanned by selected sine and cosine basis functions.