if $P\Rightarrow Q$ Then both are banach space?

Question

$X,Y$ are norm linear space and $T_n$ be a sequence of bounded linear operators from $X\to Y$ consider the two statements below

$P:\{\|T_n(x)\|\}$ is bounded for ever $n$

$Q:\{\|T_n\|\}$ is bounded for every $n$

which of the following statements is correct?

1. if $P\Rightarrow Q$ Then both are banach space

2. if $P\Rightarrow Q$ Then one of them is banach space

3. If $X$ is a banach space then $P\Rightarrow Q$

4. if $Y$ is a banach space then $P\Rightarrow Q$

I am completely lost. could anyone help me to solve?or neccessary information which will help me to solve?

Answer 1

For $1)$ and $2)$ take non-complete normed spaces, take any non-zero linear bounded operator $T:X\to Y$. To get a counterexample set $T_n=T$ for all $n\in\mathbb{N}$.

Statement $3)$ is correct by uniform boundedness principle.

For $4)$ see this counterexample.