• Title: Total Orthonormalsystem

• Series: Fourier Transform

• Chapter: Fourier Series

• YouTube-Title: Fourier Transform 9 | Total Orthonormalsystem

• Bright video: https://youtu.be/BmyIRfSgM5A

• Dark video: https://youtu.be/ma0bgdf1uwI

• Subtitle on GitHub: ft09_sub_eng.srt missing

• Timestamps (n/a)
• Subtitle in English (n/a)
• Quiz Content

Q1: Consider a vector space $V$ with an ONS $(e_i)_{i \in \mathbb{N}}$. What does it mean that the ONS is complete?

A1: For all $x \in V$ we have: $$\left| x - \sum_{k=1}^n e_k \langle e_k, x \rangle \right| \xrightarrow{ n \rightarrow \infty } 0$$

A2: For all $x \in V$ we have: $$\left| x - \sum_{k=1}^n e_k \langle e_k, x \rangle \right| \geq n$$

A3: For all $x \in V$ we have $\langle x, e_k \rangle = 0$ for all $k \in \mathbb{N}$.

A4: For all $k \in \mathbb{N}$ there is a vector $x$ with $\langle x, e_k \rangle = 0$.

Q2: Consider a vector space $V$ with an ONS $(e_i)_{i \in \mathbb{N}}$. What does it mean that the ONS is total?

A1: The linear span of $(e_i)_{i \in \mathbb{N}}$ is dense in $V$.

A2: The linear span of $(e_i)_{i \in \mathbb{N}}$ is equal to $V$.

A3: The linear span of $(e_i)_{i \in \mathbb{N}}$ is of dimension $\infty$.

A4: The linear span of $(e_i)_{i \in \mathbb{N}}$ is of finite dimension.

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