Q1: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a $k$-dimensional subspace with an ONB $\mathcal{B} = (e_1, \ldots, e_k)$. What is not correct?

A1: Each $u \in U$ can be written as a linear combination with vectors from $\mathcal{B}$.

A2: Each $u \in U$ can be written as $ u = \sum_{j=1}^k e_j \langle e_j, u\rangle$.

A3: Each $u \in U$ can be written as $ u = \sum_{j=1}^k \lambda_j e_k $, where $\lambda_j$ are called the Fourier coefficients.

A4: There is a $u \in U$ that is orthogonal to all vectors from $\mathcal{B}$.

Q2: Let $V$ be the real vector space of continuous functions on $[0,2 \pi]$ with inner product $\langle f, g\rangle = \int_0^{2\pi} f(x) g(x), dx$. Are the $\cos$ and $\sin$ function orthogonal in this space?

A1: Yes, the integral is 0.

A2: No, they are not normalized.

A3: No, the integral gives 1.

A4: The inner product is not well-defined.

Q3: Let $V$ be the real vector space of continuous functions on $[0,2 \pi]$ with inner product $\langle f, g\rangle = \int_0^{2\pi} f(x) g(x), dx$. Do $(\cos, \sin)$ form an ONS here?

A1: No!

A2: Yes!

A3: One needs more information.

Q4: Let $V$ be the real vector space of continuous functions on $[0,2 \pi]$ with inner product $\langle f, g\rangle = \int_0^{2\pi} f(x) g(x), dx$. Here $(e_1, e_2, e_3) = (\frac{1}{\sqrt{2\pi}}, \frac{1}{\sqrt{\pi}} \cos, \frac{1}{\sqrt{\pi}} \sin ) $ form an ONS. What are the Fourier coefficients for the function $f(x) = 1 + \cos(x)$?

A1: They are $\langle e_1, f\rangle = \sqrt{2 \pi} $, $\langle e_2, f\rangle = \sqrt{ \pi} $, $\langle e_3, f\rangle = 0 $.

A2: They are $\langle e_1, f\rangle = \sqrt{\pi} $, $\langle e_2, f\rangle = \sqrt{ \pi} $, $\langle e_3, f\rangle = \sqrt{ \pi} $.

A3: They are $\langle e_1, f\rangle = 1 $, $\langle e_2, f\rangle = 1 $, $\langle e_3, f\rangle = 0 $.

A4: They don’t exist.