Science & Mathematics » Mathematics » Conceptually, what is a Linear Functional?

Conceptually, what is a Linear Functional?

I can't find a good explanation of this anywhere.

Every site I see tells me, implementation-ally, what it is. For example, Wolfram MathWorld says: "A linear functional on a real vector space is a function T : V -> R, which satisfies the following properties..."

However, it just looks like a [linear] function...

I see that, in one case (I think...), a 'linear functional' could be defined as T(a,b) = 2a + b so I could say "a linear functional is a function which can be easily represented by a vector; then, applying this function to another vector is equivalent to an inner product defined as the dot product"

However, in a number of homework problems, they define something convoluted as seen in the solutions for 6.25 here:

So, what IS a linear functional at is core? There HAS to be a simple explanation as to what it is that causes the details/notation to fall into place. However, everybody seems to work the other way around: starting with the muddy details and hoping the idea pops out of their butts.

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