Spectrum of the second derivative

Discussion in 'Linear and Abstract Algebra' started by akw, Apr 9, 2022.

  1. akw

    akw

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    Hi all,

    I'm doing an online Linear Algebra course and have no way to get myself unstuck, so I'm hoping you guys might help. Here are the questions:

    Consider the subspace of smooth functions `f(x)` such that `f(a) = 0`, where `a` is some constant

    1. This does describe a subspace, doesn't it?

    To this, my answer is yes, since not all functions, and linear combinations thereof, can equal 0.

    2. Describe the full spectrum of the second derivative operator on this, more narrow, space of functions. Compare it with the spectrum of the full space of smooth functions.

    I'm stuck on this. Sure, the second derivative has the effect of zeroing constant and linear polynomial functions, but this is just me stating something I know, rather than answering the question. I'm really confused about what is being asked.

    Would be grateful for some help.
     
    akw, Apr 9, 2022
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