How to disprove $(\text B)$ and $(\text D)$?

Question

Let $X$ and $Y$ be two normed linear spaces. Let $T : X \longrightarrow Y$ be a bijective linear map with closed graph. Then which one of the following options is necessarily TRUE?

$(\text A)$ The graph of $T$ is equal to $X \times Y.$

$(\text B)$ $T^{-1}$ is continuous.

$(\text C)$ The graph of $T^{-1}$ is closed.

$(\text D)$ $T$ is continuous.

My attempt $:$ Clearly $(\text C)$ is true. Since the graph of $T$ is closed. So $X \times T(X)$ is a closed subset of $X \times Y$ and hence the graph of $T^{-1}$ i.e. $T(X) \times X$ is a closed subset of $Y \times X.$ Clearly $(\text A)$ is false . For instance let us take $X=Y= \Bbb R$ and consider the linear operator $T : X \longrightarrow Y$ defined by $T(x) = 2x,$ $x \in X.$ Then it is easy to see that $T$ is a bijective linear map with closed graph as the set of points on the line $y=2x$ is a closed subset of $\Bbb R^2.$ Now $(1,1) \in X \times Y$ but $(1,1) \notin X \times T(X).$

How do I disprove $(\text B)$ and $(\text D)$? To disprove any of these two we need to search for linear maps beween infinite dimensional normed linear spaces because any linear maps between finite dimensional normed linear spaces is bounded and hence continuous. Can anybody please help me in this regard?

Thank you ver much for your valuable time.

Answer 1

Hints for B) and D). Let $X$ be $\ell^{1}$ with its own norm and $Y$ be $\ell^{1}$ with the norm $\|.\|_2$. Let $T$ be the identity map from $X$ to $Y$. Show that this counter-example for B). For D) simply switch $X$ and $Y$.

[ Suppose $\sum |a_n| \leq C \sqrt {\sum |a_n|^{2}}$ for all $(a_n) \in \ell^{1}$. Apply the inequality to $(1,\frac 1 2, \frac 13,..., \frac 1 N,0,0,..)$ to get a contradiction].

Answer 2

(C) is indeed true but your argument is incorrect because the graph of $T$ is not $X \times T(X)$, it is $$\Gamma_T =\{(x,Tx) : x \in X\}$$

Now, the map $p : X \times Y \to Y \times X, (x,y) \mapsto (y,x)$ is a homeomorphism so $$\Gamma_{T^{-1}} = \{(y, T^{-1}y) : y \in Y\} = \{(y,x) \in Y \times X : y = Tx\} = p(\Gamma_T)$$ is closed as an image of a closed set via a homeomorphism.