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Math Help - vector space and norm space

  1. #1
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    vector space and norm space

    show that every inner product is norm.
    if so why innerproduct goes to a field F while norm only goes to R.
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  2. #2
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    Quote Originally Posted by sah_mat View Post
    show that every inner product is norm.
    if so why innerproduct goes to a field F while norm only goes to R.
    I am not exactly sure what your question is.

    If \left<~ ,  ~ \right> : V\times V\to \mathbb{C} is an inner product space then one of the conditions is that \left< x,x\right>\geq 0. The norm is define by \sqrt{\left< x,x \right>}, thus, certainly this is a function on V\times V to the non-negative real numbers.
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  3. #3
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    Quote Originally Posted by sah_mat View Post
    show that every inner product is norm.
    if so why innerproduct goes to a field F while norm only goes to R.
    You can't prove it- that is not true. And it probably was not what you were asked to prove. What is true is that any norm gives a norm. That is, given any inner product, we can use it to define a norm- but the norm and the inner product are obviously NOT the same.

    Given any inner product, u*v, say, we can define the norm of a vector v as |v*v| where "| |" is the absolute value as defined on the field F. You are asked to show that |v*v| has the defining properties of a norm.
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