Hilbert Space

Abstract

Recall our use of n-dimensional Euclidean space ℝⁿ, the set of n-vectors or n-tuples x = (x ₁,... ,x _n ) with each x _i ∈ ℝ. Here one has the ordinary Euclidean length — the norm

$$\left\| x \right\|: = {(\sum\limits_{i = 1}^n {x_i^2} )^{\frac{1}{2}}}$$

— and the inner product (or dot product) of two vectors x and y:

$$x\cdot y,: = \sum\limits_{i = 1}^n {{x_i}{y_i}} $$

.