Complex Inner-Product Spaces: True or False?

In summary, In an arbitrary complex inner-product space V, it is not always true that <αu + βv, w> = α<u, w> + β<v, w>. Option b is true if the inequality is written as |<u, v>|^2 ≤ <u, u> <v, v>. Option c is true as constants in the second argument do not get conjugated. Option d is true as multiplying a complex number by zero results in zero. The use of 'w' instead of 'v' in option b may be a typographical error.
  • #1
DanielFaraday
87
0

Homework Statement



In an arbitrary complex inner-product space V which of the following is not always true?

a. <αu + βv, w> = α<u, w> + β<v, w>
b. |<u, v>|2 ≤ <u, u> <v, w>
c. <u, αv> = α<u, v>
d. <0, u> = 0

Homework Equations


None


The Attempt at a Solution



The correct answer must be a, but I don't know what to do with b.

a. False. If the constants are complex, they will be conjugated.
b. Any ideas?
c. True. Constants in the second argument will not be conjugated.
d. True. Multiplying a complex number by zero still results in a zero.
 
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  • #2
If b) were |<u,v>|^2<=<u,u>*<v,v> it would be the Cauchy-Schwarz inequality. It's true. Is the 'w' a typo?
 
  • #3
Dick said:
If b) were |<u,v>|^2<=<u,u>*<v,v> it would be the Cauchy-Schwarz inequality. It's true. Is the 'w' a typo?

Well, my professor has a 'w' on the assignment, but you must be right.

Thanks again!
 
  • #4
Which argument gets the conjugate depends on whether it is a physics text or a math text. So consult the text.
 

What is a complex inner-product space?

A complex inner-product space is a vector space where the vectors are complex numbers and there is a defined operation for the inner product between two vectors.

How is a complex inner-product space different from a real inner-product space?

In a real inner-product space, the vectors are real numbers and the inner product operation follows the properties of the real inner product. In a complex inner-product space, the vectors are complex numbers and the inner product operation follows the properties of the complex inner product.

What are the properties of a complex inner-product space?

The properties of a complex inner-product space are linearity, conjugate symmetry, and positive definiteness. Linearity means that the inner product operation is distributive and associative with scalar multiplication. Conjugate symmetry means that the inner product of two vectors is equal to the complex conjugate of the inner product of the same two vectors in the opposite order. Positive definiteness means that the inner product of a vector with itself is always a positive real number.

How is the norm defined in a complex inner-product space?

The norm in a complex inner-product space is defined as the square root of the inner product of a vector with itself. It represents the length or magnitude of a vector.

What are some applications of complex inner-product spaces?

Complex inner-product spaces have applications in quantum mechanics, signal processing, and image processing. They are also used in the study of complex-valued functions and differential equations.

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