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The usual procedure to get the most sheen is to either use a metal polish, an acidic cleaning product, a wax product, a mild abrasive such as "Soft Scrub" or a fine steelwool. The operative word is "fine". Perhaps a short stainless steel tutorial is in order… then you will understand how to treat this material with the respect it deserves. The secret to stainless steel's strong resistance to corrosion is in its composition… it is a blend (or, more correctly, an alloy) of iron and 10% or greater chromium… with nickel added to produce a tough surface.
This alloying process produces a metal that has self-protective properties. During manufacture, the surface of the steel reacts with oxygen to form an extremely hard coating… chromium oxide… that protects the steel within. Though very durable, this coating is not invulnerable and, if penetrated, the steel will rust! Therefore, care must be taken to protect the stainless steel surface from scratches!
There is not just one stainless steel… indeed there are hundreds of formulations of this metal designed for different applications and for different environments. For example, your stainless steel kitchen sink would not keep its pristine appearance for long in sea water… most stainless steel is especially susceptible to chloride (salt)! Some types of stainless steel are non-magnetic, while others are magnetic! Some types can be welded… others not. Just understanding the varying types of stainless steel can be a life study!
Stainless steel can be cleaned and brightened with vinegar… the acidity will remove a slight amount of metal oxide. However, don't leave any acid in contact with stainless steel for a long time… it can cause permanent pitting in the surface.
Chlorine bleach should NOT be used on stainless steel. Chlorine is a very reactive chemical and can damage the steel's oxide coating.
Using a mild abrasive or steelwool pad can also help brighten up stainless steel, but there is a risk of leaving surface scratches, or "whorls", that may be aesthetically undesirable. Plus… you are wearing away the oxide! Metal polishes are a better solution. One product I am personally familiar with and highly recommend is Gel Gloss at http://www.gel-gloss.com! This product will both clean and leave a coating that will restore sheen and offer some chemical protection to the sink. Gel Gloss can also restore the stain resistance and sheen to old plastic laminate countertops that have reached their "golden" years!