Find an example of an infinite dimensional vector space for which the following conditions of matrix convergence are not equivalent.

Question

I need to find an example of linear operators $A_{n}$ and $A$ on an infinite dimensional vector space with norm $\| \cdot\|$ such that the following conditions are not equivalent:

(i) $\|A_n -A\| \to 0 \text{ as } n \to \infty$;

(ii) $|A_{n}x-Ax| \to 0 \text{ as } n\to \infty$ for each fixed $x$;

(iii) $|\langle A_{n}x,y\rangle - \langle Ax,y\rangle| \to 0 \text{ as } n\to \infty$ for each fixed $x$ and $y$.

These are equivalent for any finite dimensional vector space, though perhaps one or two examples will show that the implications do not follow for an infinite dimensional vector space.

Answer 1

Let $V=\oplus_{j=1}^\infty \Bbb R$ with the operator norm. Let $A$ be the identity function, and let $A_n$ be the identity on the first $n$ coordinates, and zero after that.

These fit the second two conditions, but the norm of the difference is always 1, so it fails the first condition.