Q1: Let $X,Y$ be two normed spaces and $T: X \rightarrow Y$ a bounded linear operator. Which is the correct condition for $T$ being a compact operator?

A1: $\overline{T[ B_1(0) ]}$ is compact.

A2: $ T[ B_1(0) ]$ is closed.

A3: $ T[ B_1(0) ]$ is compact.

A4: $ T[ B_1(0) ]$ is open.

A5: $ T[ B_1(0) ]$ is bounded.

A6: $\overline{T[ B_1(0) ]}$ is finite.

Q2: Let $X$ be a complex Banach space and $T: X \rightarrow X$ a compact operator. Which claim is, in general, not correct?

A1: $\sigma(T)$ is always a finite set.

A2: $\sigma(T)$ is always a countable set.

A3: $\sigma(T) \setminus { 0 }$ is always a countable set.

A4: $\sigma(T) \setminus { 0 }$ could be the empty set.

Q3: Let $X$ be an infinite-dimensional complex Banach space and $T: X \rightarrow X$ a compact operator. Which claim is, in general, not correct?

A1: $\sigma(T) \cap { 0 }$ is the empty set.

A2: $\sigma(T) \cap { 0 }$ is a finite set.

A3: $\sigma(T) \setminus { 0 }$ is a countable set.

A4: $\sigma(T)$ contains at least one element.