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> Let V be a vector space of finite dimension $n$ over the field K. Let $\varphi$ be a functional on $V$, and assume $\varphi\neq0$. What is the dimension of the kernel of $\varphi$? Proof?
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I think the functional $\varphi\neq 0$ means if I choose a linear function $L\_{v\_n}$ then $v\_n\neq 0$ for $n$. If we define the functional by the inner product ${v\_n,v}$, in which $v$ stands for all $v\in V$. Then the zero scalar would depend on the number of elements in $V$ that equal zero. 

**Questions**:

**1)** How can I know the kernel dimension?

**2)** Is the dimension zero due to the isomorphism between the Dual space and the Vector Space?