What have you tried so far? Can you think of results that might help here (Cauchy–Schwarz inequality perhaps, for a suitable choice of g)?Poirot said:consider C[0,2], the set of continuous functions from [0,2] to C.
The inner product is <f,g> = the integral of f(t)g(t)* from 0 to 2. show that:
sqrt(2)||f|| is greater than or equal to the magnitude of the integral of f from 0 to 2, where ||.|| is the norm of f.
Opalg said:What have you tried so far? Can you think of results that might help here (Cauchy–Schwarz inequality perhaps, for a suitable choice of g)?
An inner product space is a mathematical concept that describes a vector space with an additional structure called an inner product. The inner product assigns a scalar value to every pair of vectors in the space, which can be thought of as measuring the angle between the two vectors and their lengths.
The inner product space in C[0,2] is significant because it allows us to define an inner product on the set of continuous functions defined on the interval [0,2]. This allows us to use traditional vector space operations, such as addition and scalar multiplication, on these functions.
The integral equality in an inner product space is a property that states that the inner product of two vectors is equal to the integral of their product over the given space. In other words, it is a way of calculating the inner product using integration.
In an inner product space in C[0,2], the integral equality is satisfied because the inner product of two continuous functions is defined as the integral of their product over the interval [0,2]. This means that the inner product of two functions f and g is equal to the integral of f(x)g(x)dx from 0 to 2.
The integral equality is important because it ensures that the inner product space satisfies the axioms and properties of an inner product space. This allows us to use the inner product space to perform calculations and make mathematical conclusions about the vectors and functions within it.