The paper by Themistocles M. Rassias addresses the stability of linear mappings in Banach spaces, specifically in response to a problem posed by S. M. Ulam. The main theorem states that if a mapping \( f: E_1 \rightarrow E_2 \) satisfies a certain continuity and Lipschitz condition, then there exists a unique linear mapping \( T: E_1 \rightarrow E_2 \) that approximates \( f \) within a specified error bound. The proof involves showing that the sequence \( \{f(2^n x) / 2^n\} \) converges, and using this to construct the linear mapping \( T \). The uniqueness of \( T \) is established by demonstrating that any other linear mapping \( g \) that approximates \( f \) must be identical to \( T \). This work provides a generalized solution to Ulam’s problem, extending the results previously obtained by D. H. Hyers for the case \( p = 0 \).The paper by Themistocles M. Rassias addresses the stability of linear mappings in Banach spaces, specifically in response to a problem posed by S. M. Ulam. The main theorem states that if a mapping \( f: E_1 \rightarrow E_2 \) satisfies a certain continuity and Lipschitz condition, then there exists a unique linear mapping \( T: E_1 \rightarrow E_2 \) that approximates \( f \) within a specified error bound. The proof involves showing that the sequence \( \{f(2^n x) / 2^n\} \) converges, and using this to construct the linear mapping \( T \). The uniqueness of \( T \) is established by demonstrating that any other linear mapping \( g \) that approximates \( f \) must be identical to \( T \). This work provides a generalized solution to Ulam’s problem, extending the results previously obtained by D. H. Hyers for the case \( p = 0 \).