Demiclosedness at a point instead on a set

Question

I have been reading a few research papers for iterative techniques of approximating fixed points of nonexpansive maps. In one of the papers, I found a lemma which is as follows:-

Theorem 1: Let $C$ be a closed convex subset of a uniformly convex Banach space $E$, and $T$ a nonexpansive map on $C$. Then, $I - T$ is demiclosed at zero.

This lemma was referenced to a theorem in the book "Topics in metric fixed point theory" by Goebel and Kirk. However, in the book, the theorem is stated as follows:-

Theorem 2: Let $E$ be a uniformly convex Banach space and $C$ be a nonempty closed and convex subset of $E$. Let $T: C \rightarrow E$ be a non-expansive mapping. Then, $T$ is demiclosed on $C$.

There are two questions that come to my mind:-

1. What do we mean by demiclosedness at a point? All the definitions that I could find deal with demiclosedness on a set. In particular, we say that a map $T: C \rightarrow E$ is demiclosed if for any sequence $\left( x_n \right)$ in $C$ which converges weakly to $x \in E$ and $T \left( x_n \right) \rightarrow y$ in $E$, we have $x \in C$ and $f \left( x \right) = y$. How do we extend (or restrict) this definition to define demiclosedness at a point?

2. Are these two theorems equivalent? If so, does it mean that demiclosedness of a map on a closed and convex subset is same as saying that the demiclosed at a point?

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Edit:-

I did find out the meaning of demiclosedness at a point. It is defined as follows:-

Definition: Let $X$ be a Banach space and $T$ be a mapping with domain $D \left( T \right)$ and range $R \left( T \right)$. Then, $T$ is said to be demiclosed at a point $p \in R \left( T \right)$ if for every sequence $\left( x_n \right)_{n \in \mathbb{N}}$ in $D \left( T \right)$ that converges weakly to a point $x \in D \left( T \right)$ and the sequence $\left( Tx_n \right)_{n \in \mathbb{Z}}$ in $R \left( T \right)$ converges strongly to $p$, we have $Tx = p$.

Now, Theorem $1$ makes sense to me. However, there is still a problem in proving the theorem. While exploring more on the subject, I came across the book "Topics in Fixed Point Theory" by M. A. Khamsi. In that book, a theorem similar to Theorem $1$ is proved. However, it uses Opial's condition. Thus, now the question is: "Is the Theorem $1$ stated here still true for Banach spaces not satisfying Opial's condition?"

Answer 1

Yes this is true. Not all spaces with Opial’s property are uniformly convex and vice versa is true. The sequential space $\ell^1$ is not uniformly convex but has Opial’s property; whereas $\ell^p$ (for $p\in(1,\infty)$) has both Opial’s and uniform convexity property; however, $L^p[0,1]$ (for $p\in(1,\infty)$) has uniform convexity but does not have Opial’s property.