Let (V, ⟨⋅, ⋅⟩) be an inner product space. Prove that the inner product is a continuousfunction

To prove that the inner product is a continuous function, we need to show that for any sequences of vectors {x_n} and {y_n} in V that converge to x and y respectively, the sequence of inner products {⟨x_n, y_n⟩} converges to ⟨x, y⟩.

Step 1: Start with the definition of the inner product and its properties. The inner product is a function ⟨⋅, ⋅⟩: V × V → F (where F is the field of scalars, either R or C) that satisfies the following properties for all vectors x, y, z in V and all scalars a in F:

1. ⟨x, x⟩ ≥ 0 and ⟨x, x⟩ = 0 if and only if x = 0 (positivity)
2. ⟨x, y⟩ = ⟨y, x⟩* (symmetry, where * denotes the complex conjugate)
3. ⟨x+y, z⟩ = ⟨x, z⟩ + ⟨y, z⟩ (additivity)
4. ⟨ax, y⟩ = a⟨x, y⟩ (homogeneity)

Step 2: Use the properties of the inner product to show that |⟨x_n - x, y_n - y⟩| goes to 0 as n goes to infinity. This can be done by applying the Cauchy-Schwarz inequality and the triangle inequality, which are consequences of the properties of the inner product.

Step 3: The Cauchy-Schwarz inequality states that |⟨x, y⟩| ≤ ||x|| ||y|| for all x, y in V, where ||x|| = sqrt(⟨x, x⟩) is the norm induced by the inner product. The triangle inequality states that ||x + y|| ≤ ||x|| + ||y|| for all x, y in V.

Step 4: By applying these inequalities, we get:

|⟨x_n - x, y_n - y⟩| ≤ ||x_n - x|| ||y_n - y|| ≤ (||x_n|| + ||x||) (||y_n|| + ||y||)

Since {x_n} converges to x and {y_n} converges to y, the sequences {||x_n||} and {||y_n||} are bounded. Therefore, the right-hand side goes to 0 as n goes to infinity, which means that |⟨x_n - x, y_n - y⟩| goes to 0 as well.

Step 5: Finally, use the linearity of the inner product to write ⟨x_n, y_n⟩ as ⟨x_n - x + x, y_n - y + y⟩ = ⟨x_n - x, y_n - y⟩ + ⟨x_n - x, y⟩ + ⟨x, y_n - y⟩ + ⟨x, y⟩. As n goes to infinity, the first three terms on the right-hand side go to 0 by the argument above, so {⟨x_n, y_n⟩} converges to ⟨x, y⟩. This proves that the inner product is a continuous function.