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Can you please give a simple proof of a negative of a linear functional being a linear functional itself? Thanks in advance.

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    Can you state the definition of a linear functional? Then, can you state what it is you need to show for the proof? Perhaps doing these things will help you prove it. – 2017-01-03
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    You can either use/prove that the composition of linear functionals is again a linear functional, or check the definition for this special case. – 2017-01-03
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    What have you tried? Do you know definition of linear functional? For $f$ a functional, do you know how $-f$ is defined? – 2017-01-03
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    Just use the definition of a linear function.I.e. a function is linear if f(v+cu) = f(v)+c*f(u), where u,v are vectors and c is from field. Nothing much to do. – 2017-01-03

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It is straightforward to check that if $L_1, L_2$ are linear operators then so is $L_1 \circ L_2$.

It is straightforward to check that $n(x) = -x$ is a linear functional.

Hence given some linear $L$, the map $n \circ L$ is linear.

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    I can get behind this answer. It still insists that OP puts some effort. – 2017-01-03
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May $f$ be a linear functional. Then $f$ due it's linearity fulfils the homogeneity and additivity properties.

Let $a \in \mathbb R$ and let $v$ be a vector: $$f(a\vec v) = a f(\vec v)$$

Hint:

Is there a different approach to write $-f$ with a scalar $a$ as $a f$.