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In moving to show that the inner product, $\langle\cdot,\cdot\rangle$ is a continuous function I have the following theorem in my notes (also on page 59 of "Linear Functional Analysis", Rynne and Youngson.

"Let $X$ be an inner product space and suppose that $\{x_n\}$ and $\{y_n\}$ are convergent sequences in $X$, with $\lim_{n\to\infty}x_n=x, \lim_{n\to\infty}y_n=y$. Then,

$$\lim_{n\to\infty}\langle x_n,y_n\rangle=\langle x,y\rangle"$$

Now, I'm not too sure what this exactly means. Sure, it can be read as "the limit of the inner product of the sequences $\{x_n\},\{y_n\}$ as $n$ tends to infinity.", but what does this intuitively mean?

Although the inner product is meant to generalize the notion of the angle between two vectors (a property that the inner product satisfies but doesn't explicitly show), the inner product of two vectors, certainly in $\mathbb R^2$ and $\mathbb R^3$, is the measure of how much of one vector is going in the same direction of the other. Bearing that in mind, how does this idea of the projection of one vector onto another relate to the theorem above?

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  • $\begingroup$ It means that the inner product is a jointly continuous function of two variables on $X\times X$. $\endgroup$ – DisintegratingByParts Aug 15 '15 at 9:15
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What it means is not that hard: We have a sequence of real numbers $\langle x_n,y_n\rangle$ which converges to a real number $\langle x,y \rangle$. It doesn't matter what your interpretation of what an inner product "means" is.

So it means that the absolute value of $\left|\langle x_n,y_n \rangle - \langle x,y \rangle \right|$ gets as small as we like by picking $n$ large enough.

This can be shown using the Schwartz inequality, e.g.

The continuity itself can be formulated as the fact that if two vectors are close together, so are their respective "angles", which makes sense geometrically, I think.

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