Linear functionals
Imagine you have a big bucket full of toys. Each toy has a color, a shape, a size, and a weight. A linear functional is like someone who asks you to give them one toy at a time, and they will give you a number back based on some rule they have.
For example, one linear functional might say "give me the biggest red toy you have, and I'll tell you its size." So, you give them the biggest red toy, and they measure its size and tell you the number.
Another linear functional might say "give me all the toys that weigh less than 5 pounds, and I'll add up their sizes and tell you the total." So, you give them those toys, and they add up the sizes and give you the total number.
Linear functionals can be helpful because they let you see some important information about the toys without having to look at all of them at once. But, just like with any toy bucket, the specific toys you have will affect what information you can get from a linear functional.
For example, one linear functional might say "give me the biggest red toy you have, and I'll tell you its size." So, you give them the biggest red toy, and they measure its size and tell you the number.
Another linear functional might say "give me all the toys that weigh less than 5 pounds, and I'll add up their sizes and tell you the total." So, you give them those toys, and they add up the sizes and give you the total number.
Linear functionals can be helpful because they let you see some important information about the toys without having to look at all of them at once. But, just like with any toy bucket, the specific toys you have will affect what information you can get from a linear functional.
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