5. 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 real or complex numbers) that satisfies the following properties:

• Conjugate symmetry: ⟨x, y⟩ = ⟨y, x⟩*
• Linearity in the first argument: ⟨ax + by, z⟩ = a⟨x, z⟩ + b⟨y, z⟩ for all scalars a, b and all vectors x, y, z in V.
• Positive-definiteness: ⟨x, x⟩ ≥ 0 with equality if and only if x = 0.

Step 2: Use the properties of the inner product and the triangle inequality to show that |⟨x_n - x, y_n - y⟩| ≤ |⟨x_n - x, y_n⟩| + |⟨x, y_n - y⟩|.

Step 3: Use the Cauchy-Schwarz inequality to show that |⟨x_n - x, y_n⟩| ≤ ||x_n - x|| ||y_n|| and |⟨x, y_n - y⟩| ≤ ||x|| ||y_n - y||.

Step 4: Since {x_n} and {y_n} are convergent sequences, ||x_n - x|| → 0 and ||y_n - y|| → 0 as n → ∞. Therefore, |⟨x_n - x, y_n⟩| → 0 and |⟨x, y_n - y⟩| → 0 as n → ∞.

Step 5: Therefore, |⟨x_n - x, y_n - y⟩| → 0 as n → ∞, which means that {⟨x_n, y_n⟩} converges to ⟨x, y⟩. This proves that the inner product is a continuous function.