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Hilbert space

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1. What constitutes a Hilbert Space and how is it different (if it is) from the usual vector space?

2. What does it mean to "normalize" a function in QM vs normalize a vector in standard vector notation.

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Solution Summary

The solution gives a brief description of Hilbert Space and its connection to probability and normalization.

See Also This Related BrainMass Solution

Linear Transformations, Hilbert Space, Inner Product and Matrix Adjoint

Please see the attached file for the fully formatted problems.
Find a 5x5 matrix M>0 such that if and x(t)= then

Can we use this definition to find the adjoint of T (T is given at the end)?

This part is the additional information to solve the question above;

Let V=L2[-1,1] be the Hilbert space of functions over the time interval [-1,1] with inner product

Let P5 V be the subspace of polynomials of order 4 or less, endowed with the inner product and norm of V, and let , be its natural basis. The linear transformation S is defined as

T is defined as the restriction of S to P5.


The matrix representation of T is below;

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