Inner Product Space
An inner product space is a special type of vector space that has a mechanism for computing a version of "dot product" between vectors. An inner product is a generalized version of the dot product that can be defined in any real or complex vector space, as long as it satisfies a few conditions.
Inner products are used to help better understand vector spaces of infinite dimension and to add structure to vector spaces. Inner products are often related to a notion of "distance" within the space, due to their positive-definite property. This relation to distance allows a norm to be imposed on the space, turning it into a metric space.
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Definition
Like most elements of linear algebra, an inner product must be linear to be meaningful. It also must support some interpretation of "distance" and return meaningful values in the complex case.
A vector space \(V\) with underlying field \(\mathbb{R}\) or \(\mathbb{C}\) is known as an inner product space when equipped with an operation \(\langle \cdot, \, \cdot \rangle\) that satisfies the following:
- (Conjugate) symmetry. For all vectors \(v_1\) and \(v_2\), \(\langle v_1, \, v_2 \rangle = \overline{\langle v_2, \, v_1 \rangle}\).
- Linearity. For any scalar \(a\) and all vectors \(v_1\), \(v_2\), and \(v_3\), \(\langle a v_1, \, v_2 \rangle = a \langle v_1, \, v_2 \rangle\) and \(\langle v_1 + v_2, \, v_3 \rangle = \langle v_1, \, v_3 \rangle + \langle v_2, \, v_3 \rangle\).
- Positive-definiteness. For any vector \(v\), \(\langle v, \, v \rangle \ge 0\) and \(\langle v, \, v \rangle = 0\) implies that \(v = \textbf{0}\).
Note that in the case of a real vector space, conjugate symmetry is just symmetry. In the complex case, it is necessary for symmetry to be conjugate symmetry so that \(\langle v, \, v \rangle\) is always a real number.
What happens if conjugate symmetry is relaxed to symmetry?
Consider the vector spaces \(\mathbb{R}^2\) and \(\mathbb{C}^2\).
More generally, positive-definiteness requires that the field's underlying vector space be ordered, and since finite fields cannot be ordered, the most common (and useful) examples of ordered fields are the real and complex numbers. So it makes sense that inner product spaces are defined only over them.
The length (or norm) of a vector \(v\) is then defined to be \[\lVert v \rVert = \sqrt{\langle v, \, v \rangle}.\] Note that the length of a vector is not linear; it does satisfy linear in scalar multiplication though. For any vectors \(v_1\) and \(v_2\), the Triangle Inequality holds: \[\lVert v_1 \rVert + \lVert v_2 \rVert \ge \lVert v_1 + v_2 \rVert.\]
Examples
The simplest, and primary, example of an inner product is the common one used in \(\mathbb{R}^n\), the dot product.