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**Question 1**

**Let [itex]\mathcal{H} = \mathbb{C}^k[/itex], where [itex]\mathcal{H}[/itex] is a Hilbert space. Then let**

[tex]S = \left\{x : \sum_{i=1}^{k} |x_i| \leq 1 \right\}[/tex]

**be a subset of [itex]\mathcal{H}[/itex]. Is the subset [itex]S[/itex] open, closed or neither?**

**Question 2**

**Let [itex]\mathcal{H} = \mathbb{C}[/itex]. Then let**

[tex]S = \left\{\frac{1}{n} : n\in \mathbb{N}\right\}[/tex]

**be a subset of [itex]\mathcal{H}[/itex]. Is the subset [itex]S[/itex] open, closed or neither?**

**Question 3**

**Let [itex]\mathcal{H} = \mathbb{C}^2[/itex]. Then let**

[tex]S = \left\{(z,0) : z\in \mathbb{C}\right\}[/tex]

**be a subset of [itex]\mathcal{H}[/itex]. Is the subset [itex]S[/itex] open, closed or neither?**

**Question 4**

**Let [itex]\mathcal{H} = l^2[/itex]. Then let**

[tex]S = \left\{x : \sum_{i=1}^{\infty} |x_i|^2 < 1\right\}[/tex]

**be a subset of [itex]\mathcal{H}[/itex]. Is the subset [itex]S[/itex] open, closed or neither?**

**Question 5**

**Let [itex]\mathcal{H} = L^2([0,1])[/itex]. Then let**

[tex]S = \left\{f : f(t) \neq 0 \, \forall \, t \in [0,1]\right\}[/tex]

**be a subset of [itex]\mathcal{H}[/itex]. Is the subset [itex]S[/itex] open, closed or neither?**

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