Let X be n-dimensional vector space. Z a proper substance of X.

Let x_0 be element of X - Z (complement of Z).

Show there is a linear functional f on X such that:

f(x_0)=1 and f(x)=0 for all x element of Z.

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- May 9th 2009, 07:27 AMfrater_cpLinear functional
Let X be n-dimensional vector space. Z a proper substance of X.

Let x_0 be element of X - Z (complement of Z).

Show there is a linear functional f on X such that:

f(x_0)=1 and f(x)=0 for all x element of Z. - May 9th 2009, 04:43 PMhkito
Do you know about inner product?

- May 10th 2009, 07:05 AMHallsofIvy
Construct a basis for Z, say $\displaystyle \{u_1, u_2, ..., u_n\}$. Extend that to a basis for X: $\displaystyle \{u_1, u_2, ..., u_n, v_1, v_2, ..., v_m\}$.

Define $\displaystyle f(u_i)= 0$, $\displaystyle f(v_i)= 1$, and extend to all of X by "linearity". - May 11th 2009, 10:29 AMfrater_cpthanks for your input
how do I use the inner product?