Which of the following is true in normed linear spaces

banach-spacesfunctional-analysisoperator-theory

Let $X$ and $Y$ be normed linear spaces, and let $T : X \to Y$ be any bijective linear map with closed graph. Then which one of the following statements is TRUE?

  1. The graph of $T$ is equal to $X \times Y $

  2. $T^{-1}$ is continuous

  3. The graph of $T^{-1}$ is closed

  4. $T$ is continuous

I have no idea. I think 1 should be true as $f$ is bijective which means graph should be $X \times Y$. But the given option is 3. Why?

Best Answer

Proof of (3):

$$\text{Graph of $T^{-1}$}=\{(y,T^{-1}y)\in Y \times X: y \in Y\}$$ Let $(y_n,T^{-1}y_n) \in \text{Graph of $T^{-1}$}$ such that $(y_n,T^{-1}y_n) \longrightarrow (y,x)$. Then $y_n \to y$ and $T^{-1}y_n \to x$

Take $x_n= T^{-1}y_n$. Bijectivity of $T$ implies $Tx_n \to y$. Since graph of $T$ is closed, $Tx=y$. That is $x=T^{-1}y$ and hence the result follows

Counterexample for (1):

Already given by Arthur in the comment. The graph in this example is a straight line which is obviously not equal to $\Bbb R \times \Bbb R$

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