Def. Continuity in terms of sequences: How do I generalize to multivariate fcns?

in these terms, your z is a fixed element of the space, and your x_n and y_n are functions. your inner product (x,z) is the integral of x times z, and your norm squared of x is the integral of the square of x.for the isomorphism, x is identified with the sequence x(n)= (x, e_n), where e_n is the nth basis element, or with the function: x(t) = int x(n) e_n(t) d(t), or with the series, sum x(n) e_n.
  • #1
benorin
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Working from "Principles of Mathematical Analysis", by Walter Rudin I have gleaned the following definition of continuity of a function (which maps a subset one metric space into another):

Suppose [itex]f:E\rightarrow Y[/itex], where [itex]\left( X, d_{X}\right) \mbox{ and } \left( Y, d_{Y}\right) [/itex] are metric spaces, and [itex]E\subset X[/itex]. Let [itex]x\in E[/itex] be a limit point of E.

f is continuous at x if, and only if, for every sequence [itex]\left\{ x_{n}\right\} \rightarrow x[/itex] such that [itex]x_{n}\in E\forall n\in\mathbb{N}[/itex], we have [itex]f(x_{n})\rightarrow f(x)\mbox{ as }n\rightarrow\infty[/itex].

My question is, when generalizing the above definition to multivariate functions, the last line of the definition would include which of the following:

a. [tex]f(x_{n},y_{n})\rightarrow f(x)\mbox{ as }n\rightarrow\infty[/tex],

or

b. [tex]f(x_{n},y_{m})\rightarrow f(x)\mbox{ as }n,m\rightarrow\infty[/tex] ?

I am uncertian if I need the double limit. :rofl: DUH! I get: X is a metric space, it could be of an arbitrary dimension if need be. But suppose that it were a product of metric spaces (with different metrics,) would the question then merit investigation?
 
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  • #2
You need a sequence v_n=(x_n,y_n,z_n,...) tending to v=(x,y,z,...) where v is a vector and the metric (ie the notion of tending to) is the euclidean one, so the first of your two options, though I suspect that the second would be equivalent in some sense, since the first has to be for *all* possible sequences.
 
  • #3
benorin said:
Working from "Principles of Mathematical Analysis", by Walter Rudin I have gleaned the following definition of continuity of a function (which maps a subset one metric space into another):
Suppose [itex]f:E\rightarrow Y[/itex], where [itex]\left( X, d_{X}\right) \mbox{ and } \left( Y, d_{Y}\right) [/itex] are metric spaces, and [itex]E\subset X[/itex]. Let [itex]x\in E[/itex] be a limit point of E.
f is continuous at x if, and only if, for every sequence [itex]\left\{ x_{n}\right\} \rightarrow x[/itex] such that [itex]x_{n}\in E\forall n\in\mathbb{N}[/itex], we have [itex]f(x_{n})\rightarrow f(x)\mbox{ as }n\rightarrow\infty[/itex].
My question is, when generalizing the above definition to multivariate functions, the last line of the definition would include which of the following:
a. [tex]f(x_{n},y_{n})\rightarrow f(x)\mbox{ as }n\rightarrow\infty[/tex],
or
b. [tex]f(x_{n},y_{m})\rightarrow f(x)\mbox{ as }n,m\rightarrow\infty[/tex] ?
I am uncertian if I need the double limit. :rofl: DUH! I get: X is a metric space, it could be of an arbitrary dimension if need be. But suppose that it were a product of metric spaces (with different metrics,) would the question then merit investigation?

Neither a) nor b). Since f is a function of two variables, for f to be continuous at (x,y), it must be true that for every sequence xn converging to x and every sequence ym converging to y, the (double) sequence f(xn,ym) (with m and n going to infinity independently) must converge to f(x,y).

If you have a product of metric spaces with different metrics, then you would use the appropriate metric on each component.
 
  • #4
[itex]x_{n}\in E\forall n\in\mathbb{N}[/itex]
Ack! That's backwards! It took me a while to figure out what you meant. :tongue2: Symbolically, that's grammatically incorrect! (Though it's okay in English)

But suppose that it were a product of metric spaces (with different metrics,) would the question then merit investigation?
Sure, but I suspect that you could always do a problem by falling back on the product metric.
 
  • #5
the whole point of giving the definition for a metric space is to make you realize that dimension has nothing to do with it, but only the metric.

so in several variables, i.e. in the product of several copies of the erals, you just use any of your favorite product metrics, eucldiean, sum, max.there is no difference which one you use either, until you get to infinite dimensions.

so several variables is the same as one, unless it means infinitely many variables.

then you have to decide whether two functions are close if they are close everywhere, or their difference has small integral, or the square of their difference has small integral, or what...
 
  • #6
The context I was looking to apply it to has, in fact, infinite dimensions, namely Hilbert spaces. The particular application was proving the following:

Let H denote a Hilbert space with the inner product [itex] ( \cdot , \cdot ) : H\times H \rightarrow \mathbb{C}[/itex]. Let [itex]\parallel \cdot\parallel [/itex] denote the norm induced by the inner product. Fix [itex]z\in H[/itex]. Then the following functions are continuous mappings [itex]\forall x,y\in H[/itex]:

[tex] x\rightarrow (x,z), \\ y\rightarrow (z,y) , \\ x\rightarrow \parallel x \parallel , \mbox{ and }x,y\rightarrow (x,y) [/tex]

The first three are easy consequences of the triangle and Schartz inequalities. The proof of the fourth is from whence my question arose.

Let [itex]x_{n}\rightarrow x \mbox{ and } y_{n}\rightarrow y\mbox{ as }n\rightarrow\infty[/itex]. Then

[tex] \left| \left( x_{n}, y_{n}\right) - \left( x, y\right) \right| = \left| \left( x_{n}, y_{n}\right) - \left( x, y_{n}\right) + \left( x, y_{n}\right) - \left( x, y\right) \right| [/tex]
[tex] \leq \left| \left( x_{n}-x, y_{n}\right) \right| + \left| \left( x, y_{n}-y\right) \right| [/tex]
[tex]\leq \parallel x_{n}-x \parallel\cdot\parallel y_{n}\parallel + \parallel x \parallel\cdot\parallel y_{n}-y\parallel\rightarrow 0\mbox{ as } n\rightarrow\infty[/tex]

:uhh: so everything is continuous now, right?
 
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  • #7
hence you are using essentially the square root of the infinite sum, or integral, of the squares of the differences of the coordinates, among the examples i gave above.

and yes your calculation looks ok to me.

to connect this to my explicit examples, note that your source is using an axiomatic approach to "hilbert space" whereas all hilbert spaces are actually isomorphic to concrete function spaces, either "separable", i.e. given by square summable sequences (functions defined on Z witha discrete measure); or not, given by square integrable functions defined on a measure space.
 

1. What is the definition of continuity in terms of sequences?

The definition of continuity in terms of sequences is the property of a function where the limit of the function at a point is equal to the functional value at that point. In other words, as the input values of the function get closer and closer to a particular point, the output values also get closer and closer to the functional value at that point.

2. How is continuity related to the concept of limits?

Continuity is closely related to the concept of limits. In fact, the definition of continuity involves the limit of a function at a particular point. A function is continuous at a point if the limit of the function at that point exists and is equal to the functional value at that point.

3. Can continuity be generalized to multivariate functions?

Yes, continuity can be generalized to multivariate functions. In the multivariate case, continuity means that the function is continuous in all directions. This means that as the input values approach a particular point from any direction, the output values also approach the functional value at that point.

4. What is the difference between continuity and uniform continuity?

Continuity and uniform continuity are both properties of functions, but they have slightly different meanings. Continuity means that the function is continuous at every point in its domain, while uniform continuity means that the function is continuous over its entire domain with a constant rate of change.

5. How can I determine if a function is continuous using the epsilon-delta definition?

The epsilon-delta definition of continuity is a mathematical method used to determine if a function is continuous at a particular point. It involves setting a value for epsilon (ε) and finding a corresponding value for delta (δ) such that if the input values are within δ of the particular point, the output values will be within ε of the functional value at that point. If such a delta value can be found, the function is continuous at that point.

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