Can view adapt to implicitly reference user's default schema?
Create multiple views and/or table-valued functions. You can't override the schema without specifying it explicitly unless you build dynamic SQL to do so for you, but you can't do that in a view or a function - only a stored procedure. 

With that said, you could do that with something like this, however it will fall back to dbo (or some other schema you hard-code) if a match is not found or if you are using Windows groups for example (which prior to SQL Server 2012 do not support default schema):

CREATE PROCEDURE dbo.whatever_procedure
AS
BEGIN
  SET NOCOUNT ON;

  DECLARE @sql NVARCHAR(MAX), @s SYSNAME;

  SELECT @s = d.default_schema_name
    FROM sys.database_principals AS d
    LEFT OUTER JOIN sys.server_principals AS s
    ON d.[sid] = s.[sid]
    WHERE SUSER_SNAME() IN (d.name, s.name);

  SELECT @s = COALESCE(@s, N'dbo');

  SET @sql = N'SELECT x FROM ' + QUOTENAME(@s) 
    + '.whatever_view_name;';

  EXEC sp_executesql @sql;
END
GO


You can test this:

USE tempdb;
GO
CREATE SCHEMA foo;
GO
CREATE LOGIN flubat WITH PASSWORD='flubat',CHECK_POLICY = OFF;
GO
CREATE USER flubat FROM LOGIN flubat WITH DEFAULT_SCHEMA = foo;
GO
GRANT SELECT ON SCHEMA::foo TO flubat;
GO
CREATE VIEW dbo.whatever_view_name AS SELECT x = 'dbo';
GO
CREATE VIEW foo.whatever_view_name AS SELECT x = 'foo';
GO
-- create the above procedure
GO
GRANT EXEC ON dbo.whatever_procedure TO flubat;
GO


And then execute it as yourself and also as flubat:

EXEC dbo.whatever_procedure;
GO
EXECUTE AS USER = N'flubat';
GO
EXEC dbo.whatever_procedure;
GO
REVERT;


Results:

x 
----
dbo

x
----
foo


But like I said, this is potentially fragile, and I would lean toward either creating a view per schema, or changing the design altogether.