# Give some example of linear functional?

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

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

• Maybe $f(x,y)=x$? – Asaf Karagila Nov 29 '20 at 15:12
• okss yes got it @AsafKaragila – jasmine Nov 29 '20 at 15:25

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

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

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.

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.