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This is for a Fourier Analysis course, any help would be much appreciated, thanks


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MTH 710 - Q1.png

1. We consider vectors in C", i.e., vectors of the form u = (21, 22, . .., 2n), where 21, 22, ..., Zn are complex
numbers. Vector addition and scalar multiplication are defined in the standard way. Given two vectors
u = (21, 22, . ..,2n), V = (W1, W2, . . ., Wn), define their inner product (u, v) by the formula
(u, v) = 21W1 + 22w2 + ... + 2nWn.
(1)
We may use this definition to define the norm of a vector u by the formula
|lull = V(u, u)
(2)
Note the introduction of the complex conjugate in the definition, and also that if z; and w; are real
numbers then formula (1) is the definition of inner product that you may be more familiar with.
(a) Consult any decent linear algebra textbook to learn what is the definition of an inner product for
a (complex) vector space. Show that formula (1) satisfies the definition of an inner product on
the vector space Cn.
Show that formula (2) satisfies the definiton a norm.
(b) For the definition in formula (1) why do we use the complex conjugate instead of using the formula
(u, v) = 21W1 + 22W2+ .. . + znWn?)

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