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WIP: Hilbert/Banach space structures #27401
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@@ -373,13 +373,18 @@ The `p`-norm is defined as | |
\\|A\\|_p = \\left( \\sum_{i=1}^n | a_i | ^p \\right)^{1/p} | ||
``` | ||
with ``a_i`` the entries of ``A``, ``| a_i |`` are their [`norm`](@ref)s, and | ||
``n`` the length of ``A``. | ||
``n`` the length of ``A``. Since the `p`-norm is computed using the [`norm`](@ref)s | ||
of the entries of `A`, the `p`-norm of a vector of vectors is not compatible with | ||
the interpretation of it as a block vector in general if `p != 2`. | ||
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`p` can assume any numeric value (even though not all values produce a | ||
mathematically valid vector norm). In particular, `norm(A, Inf)` returns the largest value | ||
in `abs.(A)`, whereas `norm(A, -Inf)` returns the smallest. If `A` is a matrix and `p=2`, | ||
then this is equivalent to the Frobenius norm. | ||
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The second argument `p` is not necessarily a part of the interface for `norm`, i.e. a custom | ||
type may only implement `norm(A)` without second argument. | ||
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Use [`opnorm`](@ref) to compute the operator norm of a matrix. | ||
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# Examples | ||
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@@ -636,7 +641,10 @@ end | |
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For any iterable containers `x` and `y` (including arrays of any dimension) of numbers (or | ||
any element type for which `inner` is defined), compute the inner product (or dot product | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. "Euclidean inner product" is perhaps more descriptive. (As opposed to some weighted inner product.) Missing a close paren after "or scalar product". There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. For me and the professors who have taught me linear algebra, Euclidean inner product implies that the corresponding field are the real numbers. Especially, if a vector space over the complex numbers is considered, the inner product has been called unitary inner product. But if Euclidean inner product means just an inner product in (american) English, I can of course rename it. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. The close parenthesis is after "or scalar product, i.e. the sum of There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. The reason for the qualifier "Euclidean" or similar is to distinguish it from some other weighted inner product. But since we write that it is the sum of inner(x[i],y[i]) explicitly, I guess we don't need any other qualifiers. Yes, I would suggest having the There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. The docstring should also mention: |
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or scalar product, i.e. the sum of `inner(x[i],y[i])`) as if they were vectors. | ||
or scalar product), i.e. the sum of `inner(x[i],y[i])`, as if they were vectors. | ||
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`dot(x, y)`` and `x ⋅ y` (where `⋅` can be typed by tab-completing `\cdot` in the REPL) are | ||
synonyms for `inner(x, y)`. | ||
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# Examples | ||
```jldoctest | ||
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