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Let $i:U \to X$ be the embedding. Assume that $i^*F = E$. Then by the adjunction we have a map $F \to i_*E$ which is an embedding, since the kernel is zero on $U$, so it is a torsion sheaf, and a vector bundle doesn't have torsion subsheaves (if $X$ doesn't have embedded components). So, we have an exact triple $0 \to F \to i_*E \to G \to 0$ for some $G$, which is supported on $X \setminus U$. If $X$ satisfies and $codim_X (X\setminus U) \ge 2$, then (provided $S_2$ then condition) we have $G^* = {\mathcal Ext}^1(G,O_X) = 0$, so dualizing the sequence we see that $F^* = (i_* E)^\ast$, hence $F = (i_* E)^{\ast\ast}$, so $F$ has to be the reflexive envelope of $i_* E$. This proves the uniquencess. This also allows to construct an example of a vector bundle on $U$ which does not extend to a vector bundle on $X$.

1

Let $i:U \to X$ be the embedding. Assume that $i^*F = E$. Then by the adjunction we have a map $F \to i_*E$ which is an embedding, since the kernel is zero on $U$, so it is a torsion sheaf, and a vector bundle doesn't have torsion subsheaves (if $X$ doesn't have embedded components). So, we have an exact triple $0 \to F \to i_*E \to G \to 0$ for some $G$, which is supported on $X \setminus U$. If $X$ satisfies and $codim_X (X\setminus U) \ge 2$, then $S_2$ then $G^* = {\mathcal Ext}^1(G,O_X) = 0$, so dualizing the sequence we see that $F^* = (i_* E)^\ast$, hence $F = (i_* E)^{\ast\ast}$, so $F$ has to be the reflexive envelope of $i_* E$. This proves the uniquencess. This also allows to construct an example of a vector bundle on $U$ which does not extend to a vector bundle on $X$.