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The Cauchy-Schwarz inequality is used to bound the absolute difference of the inner products $|\langle x_n, v \rangle - \langle l, v \rangle|$ by the product of the norm of the difference $\|x_n - l\|$ and the norm of the test vector $\|v\|$. This step converts the problem of showing inner product convergence into showing that a product involving the norm goes to zero.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging strongly to $l$.; $v$ is an arbitrary element of $H$.

The Cauchy-Schwarz inequality is used to bound the absolute difference of the inner products $|\langle x_n, v \rangle - \langle l, v \rangle|$ by the product of the norm of the difference $\|x_n - l\|$ and the norm of the test vector $\|v\|$. This step converts the problem of showing inner product convergence into showing that a product involving the norm goes to zero.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging strongly to $l$.; $v$ is an arbitrary element of $H$.