I'm working on an exercise in which you need to prove the existence of a certain function, but I'm not quite sure how to do this. A previous assignment had a similar proof that I couldn't figure out either, so I'd like some assistance please!
Let $X$ and $Y$ be normed vector spaces and $T : X \to Y$ a linear map.
(a) I first want to prove that the graph $G(T)$ of $T$ is closed in $X \times Y$ if and only if for every sequence $(x_n)$ in $X$ such that $x_n \to 0$ and $(Tx_n)$ converges in $Y$, the $\lim_{n \to \infty} Tx_n = 0$.
Assume now that $X$ and $Y$ are complete and that $T$ is bounded. Let $Z$ be a Banach space for which there exists a continuous injective linear map $J : Z \to Y$. Suppose $T(X) \subset J(Z)$.
(b) I then want to show that there exists a unique linear map $\hat{T}: X \to Z$ such that $T = J \hat{T}$ and prove that $\hat{T}$ is continuous.
What I've done so far:
(a): $(\Rightarrow)$ If a sequence $(x_n)$ in $X$ converges and sequences $(y_n)$ in $Y$ converges, then the sequence $\{x_n, y_n \}$ converges in $X \times Y$.
If we define $y$ as the limit of $(Tx_n)$, then we know that $(x_n, Tx_n)$ converges in $X \times Y$. Since $(x_n, Tx_n) \in G(T)$, the graph of $T$, for every $n \in \mathbb{N}$ and $G(T)$ is closed, the limit point $(0, y) \in G(T)$.
Now, since $T$ is a linear map it sends the zero element of $X$ to the zero element of $Y$, we have y = 0.
Thus $\lim_{n \to \infty} Tx_n = 0$.
$(\Leftarrow)$ I now need to show that every sequence $(x_n)$ in $G(t)$ with has its limit point $x$ in $G(T)$. This is probably very simple, but I don't see how the statement in the exercise can help me prove this...
(b) As I've said earlier, I'm completely stumped on how to do this...
I thought about just assuming that there is linear map with these properties and trying to show that $T$ is then still linear, but I don't think that that is the correct way to do it...
I know that this is homework and that I've only done a small part of the exercise, so I would be happy with even the smallest hint. This type of question is bound to come up again, so I'd really like to know a general way to solve them.
Thanks in advance for any replies!