Introduction:
In this question, we are given two non-zero functionals, f and g, on a real vector space V. We need to determine the nature of the function v under certain conditions.

Solution:
To determine the nature of the function v, let's analyze the given options one by one.

(a) v is also linear:
For a function to be linear, it must satisfy two properties: additivity and homogeneity.

Additivity: v(x + y) = v(x) + v(y)
Homogeneity: v(cx) = cv(x)

Let's check if v satisfies these properties.

Since f and g are functionals on V, they are linear. Therefore, we have:

f(x + y) = f(x) + f(y) (Property 1)
g(x + y) = g(x) + g(y) (Property 2)
f(cx) = cf(x) (Property 3)
g(cx) = cg(x) (Property 4)

Now, let's consider v(x + y) and v(cx):

v(x + y) = f(x + y) + g(x + y) (Property 5)
= f(x) + f(y) + g(x) + g(y) (Properties 1 and 2)
= (f(x) + g(x)) + (f(y) + g(y))

v(cx) = f(cx) + g(cx) (Property 6)
= cf(x) + cg(x) (Properties 3 and 4)
= c(f(x) + g(x))

From the above equations, we can observe that v satisfies both additivity and homogeneity properties. Hence, v is also linear.

Therefore, option (a) is correct.

(b) v is not well-defined:
A function is well-defined if it gives a unique output for each input. Since we are not given any specific conditions or restrictions on f and g, we cannot determine whether v is well-defined or not based on the given information. Therefore, option (b) is not a conclusive answer.

(c) v is a quadratic form:
A quadratic form is a function of the form Q(x) = x^T A x, where A is a symmetric matrix and x is a vector.

Based on the given information, we cannot directly conclude that v is a quadratic form. Therefore, option (c) is incorrect.

Conclusion:
From the analysis, we can conclude that the correct option is (a) v is also linear.