Give some example of linear functional?

Question

Give some example of linear functional ?

My attempt : I know the definition of linear functional

linear functional on $V$ is a function $ f : V \to F$ such that

$f(a_1 v_1 + a_2v_2)=a_1f(v_1) + a_2f(v_1)$ for all $ a_1,a_2 \in F$ and all $v_1,v_2 \in V$

Here i don't know how to find the example of linear functional

But i can find the example of linear transformation take $f : \mathbb{R}^3 \to \mathbb{R}^3$ defined

$f(x,y,z)= (x+y,y+z,z+x)$

Answer 1

An example that is close to the example you have of a linear transformation:

$$f(x,y,z)=x+y$$

This is a linear functional on $\mathbb{R}^3$ or, more generally, $F^3$ for any field $F$.

A much more interesting example of a linear functional is this: take as your vector space any space of nice functions on the interval $[0,1]$, for example the space of continuous functions or the space of polynomials or (if you prefer a finite dimensional space) the space of polynomials of degree at most $20$. Now you have a linear functional

$$\phi(f)=\int_0^1 f(x)\,dx$$.

Answer 2

Let $\gamma_i\in C[a,b]$ and $V=C^n [a,b]$ and $F:V\to\Bbb{R}$.Then, $$F(v)=\int_a^b [\gamma_0(x)v(x)+\gamma_1(x)h'(x)+...+\gamma_nh^{n}(x)]dx$$ is a linear functional.

Answer 3

Common examples are integrals. Another (apparently different) example is differentiation. Take as your space the collection of (once) differentiable functions on some open interval, say $(0,1)$. Then use linearity of the derivative: $$ \frac{\mathrm{d}}{\mathrm{d}x} \left( a f(x) + b g(x) \right) = a \frac{\mathrm{d}f(x)}{\mathrm{d}x} + b \frac{\mathrm{d}g(x)}{\mathrm{d}x} \text{.} $$

If you demand finite dimensionality, use polynomials of bounded degree.