Foundations — Inner product spaces — dot product generalization

This page is the ground floor. The parent note uses roughly a dozen symbols and quietly assumes you already picture them. Here we earn every single one, in an order where each brick rests on the one below it. Nothing is used before it is drawn.

0. What is a "vector" here, really?

Before any symbol appears, fix the picture.

• $v_1$ = how far along the horizontal axis (the "right" step).
• $v_2$ = how far along the vertical axis (the "up" step).

Why the topic needs this: the whole subject is about doing geometry to arrows. But the punchline of the parent note is that "arrow" can later mean a polynomial or a sound wave. So we start with the arrow you can literally see, then generalise. See Dot Product for where these components come from.

1. The symbols $\mathbf{u},\mathbf{v},\mathbf{w}$ and $0$

Why we need $0$ specially: the third axiom of an inner product singles it out — it is the only arrow allowed to have zero length. Every other arrow must have positive length. Keep this arrow in mind; it is the "degenerate case" the axioms are built to handle.

2. Scaling: the symbol $c\mathbf{v}$

Why the topic needs this: the linearity axiom is a promise about how the machine reacts to stretching. To even state that axiom you must first know what "stretch by $c$" looks like — that is the picture above.

3. Adding arrows: $\mathbf{u}+\mathbf{v}$

Why the topic needs this: the triangle inequality $\Vert \mathbf{u}+\mathbf{v}\Vert \le \Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert$ is literally a statement about this tip-to-tail triangle: the direct path is never longer than the detour. You cannot read that inequality without seeing this picture.

4. The dot product $\mathbf{u}\cdot \mathbf{v}$ — the original magic ruler

Now the star of the show.

Why this single operation matters: the whole parent topic rests on the observation that this one number secretly encodes three geometric facts — length, angle, perpendicularity. The next three symbols pull each fact out.

5. Norm $\Vert \mathbf{v}\Vert$ — length from the dot product

Why the square root and not something else? Because $v_1^2+v_2^2$ is the squared hypotenuse of the right triangle with legs $v_1$ and $v_2$. To get the actual length we must undo the square — that is precisely what  ​ does. This is the tool that answers the question "how long is this arrow?" More in Norms and Distance.

6. The angle $\theta$ and $\cos \theta$

Why $\cos$ and not $\sin$ or $\tan$? We want a number that is $+1$ when arrows align, $0$ when perpendicular, $-1$ when opposite. That is exactly the behaviour of $\cos$: $\cos 0^{\circ }=1$, $\cos 90^{\circ }=0$, $\cos 180^{\circ }=-1$. No other basic trig function fits all three checkpoints, so cosine is the natural choice.

7. Orthogonality: the symbol $\perp$

Why this is the cleanest of the three jobs: length and angle need a square root and a division. Perpendicularity needs nothing — just "is the number zero?". That simplicity is why Orthogonality and Gram-Schmidt, Orthogonal Projections, and Fourier Series all lean on it so heavily.

8. The generalised bracket $⟨\mathbf{u},\mathbf{v}⟩$

Here is the leap the parent note makes.

Once you have any $⟨,⟩$, you copy §5–§7 word for word: $\Vert \mathbf{v}\Vert =\sqrt{⟨\mathbf{v},\mathbf{v}⟩},\cos \theta =\frac{⟨\mathbf{u},\mathbf{v}⟩}{\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert },\mathbf{u}\perp \mathbf{v}⟺⟨\mathbf{u},\mathbf{v}⟩=0$

9. The leftover symbols

Why the integral is the right generalisation: a vector $(u_1,\dots ,u_n)$ is a list of numbers indexed by $i$. A function $f(x)$ is a list of numbers indexed by $x$ — but $x$ runs continuously. The dot product ${\sum }_i{u}_i{v}_i$ therefore becomes $\int f(x)g(x)dx$: multiply matching values, then add them all up. Same idea, continuous bookkeeping.

How these foundations feed the topic

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