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This is without question the best food for baby humans. It is cheap, safe, and nutritionally perfect for healthy term babies. It confers significant protection against infectious disease.
Does the mother's diet influence baby? Maternal diet gets blamed all the time for baby's ills, but I have found this to be pretty rare in practice.
The issue of spicy foods is pretty easy. Most of the world's mothers breastfeed, because they are poor. Their diets are fairly simple, and they have developed some pretty spicy cuisines to liven up their diets. And yet their babies are generally very happy. Some of the essential oils from foods will get through in the breast milk of mammals - e.g. onions. I have yet to see a mother-infant pair in whom this was a major issue; usually the worst that could happen is that the infant would wrinkle his nose at the funny taste. They are usually so hungry and happy to nurse they just don't seem to mind.
As for gas-producing foods, forget it. Gas is swallowed air, period. Beans give people gas, for example, because microorganisms in the human gut ferment certain sugars in the beans to produce gases. These gases DO NOT get in the mother's bloodstream to be secreted in the milk. "Gee, honey, your milk is awfully foamy tonight." "Yes, it must have been those beans I ate..."
It is thought that sometimes fragments of cow milk protein enter the mother's bloodstream, and are secreted in the mother's milk, causing symptoms of milk allergy in the baby. This must be somewhat rare, because of the chain of circumstance that has to occur: mother absorbs peptides, baby is sensitized to cow milk in utero or shortly thereafter. I've tried milk-free diets for colicy crying before, but it hasn't helped very often.
For sensitivities to food in general, I always counsel mothers to try the supposedly offending food several times to see if the effect on the baby is reproducible at least three times. (The same goes for milk free diets in the mother.) See breastfeeding.
Whether vitamins are necessary for every breastfeeding baby is probably debatable. I routinely prescribe them, but some "authorities" think they aren't necessary. I think they are pretty cheap insurance. The iron in human milk is very well absorbed - there is a special protein (lactoferrin) in breast milk that binds up the iron in the gut and delivers it into the baby's system with very little waste. Vitamin D intake is more problematic, and there are cases of rickets (vitamin D deficiency) reported in breast fed babies who were not supplemented.