Past exam of the mathematical methods I course of the University of Oxford / 2021 / 9 / b / Solution

First check that the proposed space is closed under the stated operations: $|\alpha a_i+b_i{|}^2\le 2{\alpha }^2{a}_i^2+2b_i^2$, and summing gives a finite result. For any finite $N$, the Cauchy-Schwarz inequality givesThe left side is an increasing, bounded sequence of real numbers. Its limit exists, proving absolute convergence of ${\sum }_i{a}_i{b}_i$. This justifies termwise addition and scalar multiplication in the defining sum. Symmetry and bilinearity follow, while $⟨A,A⟩={\sum }_i{a}_i^2\ge 0$, with equality only when every $a_i=0$. Thus it is a real inner product on the real l2 sequence space.