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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.

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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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