Every vector space over R or C can be made into an inner product space

• Feb 15th 2011, 11:16 AM
mysteriouspoet3000
Every vector space over R or C can be made into an inner product space
Let V be a vector space over F = R or C.

For the case when dim V over F is finite, say n, if we let B be a basis for V,
a vector v in V has a corresponding coordinate vector with respect to the ordered basis B. Denote this representation as [v] = (a_1 ... a_n).

It is not difficult to check that
f(v,w) = sum of (conjugate(a_i) times b_i) where [v] = (a_1 ... a_n)
and [w]=(b_1 ... b_n) will work as an inner product.
Hence, V is an inner product space over F.

But I don't have an idea on what inner product to consider for the case when V is infinite dimensional over F= R or C.
• Feb 15th 2011, 02:58 PM
tonio
Quote:

Originally Posted by mysteriouspoet3000
Let V be a vector space over F = R or C.

For the case when dim V over F is finite, say n, if we let B be a basis for V,
a vector v in V has a corresponding coordinate vector with respect to the ordered basis B. Denote this representation as [v] = (a_1 ... a_n).

It is not difficult to check that
f(v,w) = sum of (conjugate(a_i) times b_i) where [v] = (a_1 ... a_n)
and [w]=(b_1 ... b_n) will work as an inner product.
Hence, V is an inner product space over F.

But I don't have an idea on what inner product to consider for the case when V is infinite dimensional over F= R or C.

Just generalize the finite dimensional case: choose a basis for the vector space (AC is needed here for

inf. dim. cases) and define the inner product as before, taking into account that any element in the v.s. is

a FINITE lin. comb. of some elements in the basis.

Tonio
• Apr 1st 2011, 02:40 AM
mysteriouspoet3000
Quote:

Originally Posted by tonio
Just generalize the finite dimensional case: choose a basis for the vector space (AC is needed here for

inf. dim. cases) and define the inner product as before, taking into account that any element in the v.s. is

a FINITE lin. comb. of some elements in the basis.

Tonio

Thank you very much! Great help. :-)