If $x_k\to x$ and $y_k\to y$, then $\inner{x_k}{y_k}\to\inner{x}{y}$.Suggest a slogan
Let \(H\) be an inner product space with inner product \(\inner{\cdot}{\cdot}\) and norm \(\norm{x}\coloneqq\inner{x}{x}^{1/2}\). If \(x_k\to x\) and \(y_k\to y\) in \(H\), then \(\inner{x_k}{y_k} \to \inner{x}{y}\).
Add and subtract \(\inner{x}{y_k}\): \[ \abs{\inner{x_k}{y_k} - \inner{x}{y} } = \abs{\inner{x_k - x}{y_k} + \inner{x}{y_k - y} } \leq \norm{x_k - x}\norm{y_k} + \norm{x}\norm{y_k - y} \] by the triangle inequality and the Cauchy–Schwarz inequality. Since \(y_k \to y\), the sequence \((\norm{y_k})\) is bounded, and \(\norm{x_k - x} \to 0\), so the first term tends to \(0\). Since \(\norm{y_k - y}\to 0\), the second term tends to \(0\).