The dot product is defined for any $\mathbf{u,v}\in\mathbb{R}^n$ as,

$$

\mathbf{u} \cdot \mathbf{v} =\mathbf{u}^{\mathsf{T}} \mathbf{v}=\sum_{i=1}^{n} u_{i} v_{i}=u_{1} v_{1}+\cdots+u_{n} v_{n}

$$

Recall the geometric definition for $\mathbf{u,v}\in\mathbb{R}^{n}$ when $1\leq n\leq3$

$$

\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\|\|\mathbf{v}\|\cos[\measuredangle(\mathbf{u},\mathbf{v})]

$$

In 1D, 2D, and 3D, the oriented angle measured between two vectors makes sense.

From this I have two questions:

(1) Does the geometric definition extend to cases where $n\geq4$? I cannot imagine $\measuredangle(\mathbf{u},\mathbf{v})$ making sense in higher dimensions.

(2) Is the dot product always defined with the 2-norm? Would it still make sense to use any other $p$-norm? Or a general norm?

Extra question (if it makes sense): What about infinite-dimensional spaces?

Thank you for the insight.