In this paper, we will discuss the weak stability of ε-isometries on certain Banach spaces. Let f: X → Y be a standard ε-isometry. If Y^* is strictly convex, then for any x^*∈X^*, there is φ∈Y^* that satisfies ‖φ‖ ≡r=‖x^* ‖, such that |〈x^*,x〉-〈φ,f(x)〉|≤2rε,x∈X. Also, we show that if X and Y are both L_P spaces (1<p<∞), f: X→Y is a standard ε-isometry, then there exists a linear operator T: Y→X with norm 1 such that ‖Tf(x)-x‖≤2ε,∀x∈X.

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