Definition 1 (Normed Linear Space). Given a field k which has an absolute value function defined on it, a normed linear space is a vector space V over k equipped with a norm map || · || : V → ℝ which has the following properties:

1. ||x|| ≥ 0 for all x ∈ V.

2. ||x + y|| ≤ ||x|| + ||y|| for all x, y ∈ V (triangle inequality).

3. ||αx|| = |α| ||x|| for x ∈ V and α ∈ k.

4. ||x|| = 0 only if x = 0.

Definition 2 (Bounded Linear Functional). A bounded linear functional on V (a vector space over k) is a linear transformation T : V → k such that

sup_{x≠0} \frac{|T (x)|}{||x||} < ∞.

This sup is called the norm of the linear functional.

Definition 3 (Dual Space). The set of all bounded linear functionals on V is denoted V^{∗} and called the dual space of V when given the obvious vector space structure and the normed linear space structure arising from the norms of the linear functionals.

Exercise 1. Prove that for finite-dimensional V, V^{∗} is isomorphic to V as a normed linear space.

Exercise 2. Let ℓ^{1} = {(x_{i}) | x_{i} ∈ ℝ, ∑ |x_{i}| < ∞}. Let ℓ^{∞} = {(x_{i}) | x_{i} ∈ ℝ, sup |x_{i}| < ∞}. Prove that (ℓ^{1})^{∗} is isomorphic to ℓ^{∞}, that ℓ^{1} has a dense countable subset, and that ℓ^{∞} does not have a dense countable subset. So for infinite-dimensional V, V^{∗} may not be isomorphic to V.

Definition 4 (Banach Space). A Banach space is a normed linear space which is complete in the metric induced by the norm.

Definition 5 (Bilinear Form). A bilinear form B on a vector space V over a field k is a map B : V ×V → k which satisfies the following properties:

B(x, αy + βz) = αB(x, y) + βB(x, z) and B(αx + βy, z) = αB(x, z) + βB(y, z).

Definition 6 (Hermetian Form). A Hermetian form H on a vector space V over ℂ is a map B : V ×V → k which satisfies the following properties:

B(αx + βy, z) = αB(x, z) + βB(y, z)
B(x, αy + βz) = \overline{α}B(x, y) + \overline{β}B(x, z)
B(x, y) = \overline{B(y, x)}

Definition 7 (Inner Product). On a vector space over ℝ, an inner product is a positive definite symmetric form. On a vector space over ℂ, an inner product is a positive definite Hermetian form. In either case, the inner product B gives a normed linear space structure by setting ||v|| = \sqrt{B(v, v)}.

Definition 8 (Hilbert Space). A Hilbert space is a vector space over ℂ equipped with an inner product which is a Banach space with respect to the norm induced by the inner product.

References

Artin, M. Algebra. (Bilinear forms.) Rudin, W. Real & Complex Analysis. (Everything else.)