Inner Product Spaces and Cauchy Schwarz, Real Analysis II
In this lecture, we define a real inner product space, using the familiar Euclidean space R^n as a reference. We'll start with the definition of an inner product, which generalizes the dot product by maintaining key properties like positivity, non-degeneracy, multiplicativity, distributivity, and symmetry.
(MA 426 Real Analysis II, Lecture 2)
After discussing these properties in detail, we'll explore an example of an inner product space that isn't Euclidean space. We'll consider the set of continuous functions on the interval [0,1], defining an inner product using the integral of the product of two functions over this interval. We'll verify that this example satisfies all the necessary properties of an inner product space.
Finally, we'll wrap up by proving the Cauchy-Schwarz inequality. This involves demonstrating that for any two vectors in a real inner product space, the square of their inner product is less than or equal to the product of the inner products of each vector with itself.
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