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In a metric space, a set is open if for any element of the set we can find an open ball about it that is contained in the set. Well for the singletons in the discrete space, every other element is said to have a distance away of 1. So we can make a ball about the singleton of radius 1/2 ... this ball just equals that singleton since it contains only that element. So it is contained in the set. Thus the singleton set is open.
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Prove that Hilbert Space is a Metric Space?
The question doesn't make sense, or alternatively it is true by definition. A Hilbert Space is a complete inner product space - complete in the metric induced by the norm defined by the inner product over the space. In other words an inner product space is a vector space with an inner product defined on it. An inner product then defines a norm on the space, and every norm on a space induces a metric. A Hilbert Space is thus also a complete metric space, simply where the metric is induced by the inner product.
What do yo mean by Discrete mathematics?
Discrete Mathematics is mathematics that deals with discrete objects and operations, often using computable and/or iterative methods. It is usually opposed to continuous mathematics (e.g. classical calculus). Discreteness here refers to a property of subjects of discourse. Some collection of things is called discrete if these things are distinguishable and not continuously transformable into each other. An example would be the collection of natural numbers, but not the real numbers. In topology, a space is called discrete if every subset is open. In constructivism, a set is called discrete if equality of two elements is always decidable.
When we plot all the points that satisfy an equation or inequality what do we do?
We identify a set of points in the relevant space which are part of the solution set of the equation or inequality. The space may have any number of dimensions, the solution set may be contiguous or in discrete "blobs".
What comes at the end of every whole number?
Nothing: a blank space.
How metric cubic of palm kernel shell for 1 MT?
Oh, dude, you're hitting me with that metric lingo! Alright, so 1 metric ton is equal to 1,000 kilograms. If we're talking about palm kernel shells, they have a density of about 0.6 - 0.7 metric cubic meters per ton. So, for 1 metric ton, you're looking at around 0.6 - 0.7 metric cubic meters of palm kernel shells. But hey, who's really counting, right?
